id	sid	tid	token	lemma	pos
cana-2319	1	1	communications	communication	NOUN
cana-2319	1	2	on	on	ADP
cana-2319	1	3	applied	apply	VERB
cana-2319	1	4	nonlinear	nonlinear	ADJ
cana-2319	1	5	analysis	analysis	NOUN
cana-2319	1	6	issn	issn	NOUN
cana-2319	1	7	:	:	PUNCT
cana-2319	1	8	1074	1074	NUM
cana-2319	1	9	-	-	PUNCT
cana-2319	1	10	133x	133x	NUM
cana-2319	1	11	vol	vol	NOUN
cana-2319	1	12	32	32	NUM
cana-2319	1	13	no	no	NOUN
cana-2319	1	14	.	.	PUNCT
cana-2319	2	1	1s	1s	NUM
cana-2319	2	2	(	(	PUNCT
cana-2319	2	3	2025	2025	NUM
cana-2319	2	4	)	)	PUNCT
cana-2319	2	5	552	552	NUM
cana-2319	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2319	2	7	a	a	DET
cana-2319	2	8	connection	connection	NOUN
cana-2319	2	9	between	between	ADP
cana-2319	2	10	knot	knot	NOUN
cana-2319	2	11	theory	theory	NOUN
cana-2319	2	12	and	and	CCONJ
cana-2319	2	13	linear	linear	PROPN
cana-2319	2	14	algebra	algebra	PROPN
cana-2319	2	15	ayman	ayman	PROPN
cana-2319	2	16	abouzaid	abouzaid	PROPN
cana-2319	2	17	ezelarab	ezelarab	PROPN
cana-2319	2	18	aboufattoum	aboufattoum	PROPN
cana-2319	2	19	department	department	PROPN
cana-2319	2	20	of	of	ADP
cana-2319	2	21	mathematics	mathematic	NOUN
cana-2319	2	22	,	,	PUNCT
cana-2319	2	23	faculty	faculty	NOUN
cana-2319	2	24	of	of	ADP
cana-2319	2	25	science	science	NOUN
cana-2319	2	26	,	,	PUNCT
cana-2319	2	27	kuwait	kuwait	PROPN
cana-2319	2	28	university	university	PROPN
cana-2319	2	29	,	,	PUNCT
cana-2319	2	30	p.	p.	PROPN
cana-2319	2	31	o.	o.	PROPN
cana-2319	2	32	box	box	PROPN
cana-2319	2	33	5969	5969	NUM
cana-2319	2	34	,	,	PUNCT
cana-2319	2	35	safat	safat	NOUN
cana-2319	2	36	13060	13060	NUM
cana-2319	2	37	state	state	NOUN
cana-2319	2	38	of	of	ADP
cana-2319	2	39	kuwait	kuwait	PROPN
cana-2319	2	40	.	.	PUNCT
cana-2319	3	1	e	e	X
cana-2319	3	2	-	-	NOUN
cana-2319	3	3	mail	mail	NOUN
cana-2319	3	4	:	:	PUNCT
cana-2319	3	5	ayman.aboufattoum@ku.edu.kw	ayman.aboufattoum@ku.edu.kw	ADJ
cana-2319	3	6	article	article	NOUN
cana-2319	3	7	history	history	NOUN
cana-2319	3	8	:	:	PUNCT
cana-2319	3	9	received	receive	VERB
cana-2319	3	10	:	:	PUNCT
cana-2319	3	11	29	29	NUM
cana-2319	3	12	-	-	SYM
cana-2319	3	13	08	08	NUM
cana-2319	3	14	-	-	PUNCT
cana-2319	3	15	2024	2024	NUM
cana-2319	3	16	revised	revise	VERB
cana-2319	3	17	:	:	PUNCT
cana-2319	3	18	15	15	NUM
cana-2319	3	19	-	-	SYM
cana-2319	3	20	10	10	NUM
cana-2319	3	21	-	-	PUNCT
cana-2319	3	22	2024	2024	NUM
cana-2319	3	23	accepted	accept	VERB
cana-2319	3	24	:	:	PUNCT
cana-2319	3	25	01	01	NUM
cana-2319	3	26	-	-	SYM
cana-2319	3	27	11	11	NUM
cana-2319	3	28	-	-	PUNCT
cana-2319	3	29	2024	2024	NUM
cana-2319	3	30	abstract	abstract	NOUN
cana-2319	3	31	:	:	PUNCT
cana-2319	3	32	in	in	ADP
cana-2319	3	33	knot	knot	PROPN
cana-2319	3	34	theory	theory	NOUN
cana-2319	3	35	literature	literature	NOUN
cana-2319	3	36	,	,	PUNCT
cana-2319	3	37	a	a	DET
cana-2319	3	38	lot	lot	NOUN
cana-2319	3	39	of	of	ADP
cana-2319	3	40	references	reference	NOUN
cana-2319	3	41	treated	treat	VERB
cana-2319	3	42	the	the	DET
cana-2319	3	43	topic	topic	NOUN
cana-2319	3	44	of	of	ADP
cana-2319	3	45	computing	compute	VERB
cana-2319	3	46	the	the	DET
cana-2319	3	47	alexander	alexander	PROPN
cana-2319	3	48	polynomial	polynomial	PROPN
cana-2319	3	49	and	and	CCONJ
cana-2319	3	50	proved	prove	VERB
cana-2319	3	51	that	that	SCONJ
cana-2319	3	52	the	the	DET
cana-2319	3	53	alexander	alexander	PROPN
cana-2319	3	54	polynomial	polynomial	PROPN
cana-2319	3	55	is	be	AUX
cana-2319	3	56	a	a	DET
cana-2319	3	57	knot	knot	NOUN
cana-2319	3	58	invariant	invariant	ADJ
cana-2319	3	59	.	.	PUNCT
cana-2319	4	1	in	in	ADP
cana-2319	4	2	this	this	DET
cana-2319	4	3	paper	paper	NOUN
cana-2319	4	4	,	,	PUNCT
cana-2319	4	5	we	we	PRON
cana-2319	4	6	concentrate	concentrate	VERB
cana-2319	4	7	on	on	ADP
cana-2319	4	8	an	an	DET
cana-2319	4	9	unconventional	unconventional	ADJ
cana-2319	4	10	connection	connection	NOUN
cana-2319	4	11	between	between	ADP
cana-2319	4	12	knot	knot	NOUN
cana-2319	4	13	theory	theory	NOUN
cana-2319	4	14	and	and	CCONJ
cana-2319	4	15	linear	linear	ADJ
cana-2319	4	16	algebra	algebra	NOUN
cana-2319	4	17	in	in	ADP
cana-2319	4	18	computing	compute	VERB
cana-2319	4	19	the	the	DET
cana-2319	4	20	alexander	alexander	PROPN
cana-2319	4	21	polynomial	polynomial	PROPN
cana-2319	4	22	and	and	CCONJ
cana-2319	4	23	introduce	introduce	VERB
cana-2319	4	24	an	an	DET
cana-2319	4	25	approach	approach	NOUN
cana-2319	4	26	depending	depend	VERB
cana-2319	4	27	linear	linear	NOUN
cana-2319	4	28	algebra	algebra	NOUN
cana-2319	4	29	tools	tool	NOUN
cana-2319	4	30	to	to	PART
cana-2319	4	31	show	show	VERB
cana-2319	4	32	that	that	SCONJ
cana-2319	4	33	the	the	DET
cana-2319	4	34	alexander	alexander	PROPN
cana-2319	4	35	polynomial	polynomial	PROPN
cana-2319	4	36	is	be	AUX
cana-2319	4	37	a	a	DET
cana-2319	4	38	knot	knot	NOUN
cana-2319	4	39	invariant	invariant	ADJ
cana-2319	4	40	.	.	PUNCT
cana-2319	5	1	also	also	ADV
cana-2319	5	2	,	,	PUNCT
cana-2319	5	3	we	we	PRON
cana-2319	5	4	connect	connect	VERB
cana-2319	5	5	this	this	DET
cana-2319	5	6	approach	approach	NOUN
cana-2319	5	7	to	to	ADP
cana-2319	5	8	the	the	DET
cana-2319	5	9	way	way	NOUN
cana-2319	5	10	of	of	ADP
cana-2319	5	11	fox	fox	NOUN
cana-2319	5	12	coloring	color	VERB
cana-2319	5	13	to	to	PART
cana-2319	5	14	suggest	suggest	VERB
cana-2319	5	15	another	another	DET
cana-2319	5	16	way	way	NOUN
cana-2319	5	17	to	to	PART
cana-2319	5	18	prove	prove	VERB
cana-2319	5	19	that	that	SCONJ
cana-2319	5	20	the	the	DET
cana-2319	5	21	3	3	NUM
cana-2319	5	22	-	-	PUNCT
cana-2319	5	23	coloring	coloring	NOUN
cana-2319	5	24	is	be	AUX
cana-2319	5	25	a	a	DET
cana-2319	5	26	knot	knot	NOUN
cana-2319	5	27	invariant	invariant	ADJ
cana-2319	5	28	and	and	CCONJ
cana-2319	5	29	confirm	confirm	VERB
cana-2319	5	30	what	what	PRON
cana-2319	5	31	is	be	AUX
cana-2319	5	32	already	already	ADV
cana-2319	5	33	known	know	VERB
cana-2319	5	34	about	about	ADP
cana-2319	5	35	the	the	DET
cana-2319	5	36	3coloring	3coloring	NUM
cana-2319	5	37	of	of	ADP
cana-2319	5	38	the	the	DET
cana-2319	5	39	trefoil	trefoil	NOUN
cana-2319	5	40	and	and	CCONJ
cana-2319	5	41	the	the	DET
cana-2319	5	42	figure	figure	NOUN
cana-2319	5	43	eight	eight	NUM
cana-2319	5	44	knots	knot	NOUN
cana-2319	5	45	..	..	PUNCT
cana-2319	6	1	keywords	keyword	NOUN
cana-2319	6	2	:	:	PUNCT
cana-2319	6	3	knot	knot	NOUN
cana-2319	6	4	theory	theory	NOUN
cana-2319	6	5	,	,	PUNCT
cana-2319	6	6	linear	linear	PROPN
cana-2319	6	7	algebra	algebra	NOUN
cana-2319	6	8	.	.	PUNCT
cana-2319	7	1	1	1	X
cana-2319	7	2	.	.	X
cana-2319	7	3	introduction	introduction	NOUN
cana-2319	7	4	alexander	alexander	PROPN
cana-2319	7	5	polynomial	polynomial	PROPN
cana-2319	7	6	is	be	AUX
cana-2319	7	7	the	the	DET
cana-2319	7	8	first	first	ADJ
cana-2319	7	9	polynomial	polynomial	ADJ
cana-2319	7	10	knot	knot	NOUN
cana-2319	7	11	invariant	invariant	PROPN
cana-2319	7	12	discovered	discover	VERB
cana-2319	7	13	.	.	PUNCT
cana-2319	8	1	it	it	PRON
cana-2319	8	2	is	be	AUX
cana-2319	8	3	a	a	DET
cana-2319	8	4	well	well	ADV
cana-2319	8	5	understood	understand	VERB
cana-2319	8	6	classical	classical	ADJ
cana-2319	8	7	knot	knot	NOUN
cana-2319	8	8	invariant	invariant	ADJ
cana-2319	8	9	discovered	discover	VERB
cana-2319	8	10	by	by	ADP
cana-2319	8	11	j.	j.	PROPN
cana-2319	8	12	w.	w.	PROPN
cana-2319	8	13	alexander	alexander	PROPN
cana-2319	8	14	in	in	ADP
cana-2319	8	15	1927	1927	NUM
cana-2319	8	16	.	.	PUNCT
cana-2319	9	1	there	there	PRON
cana-2319	9	2	are	be	VERB
cana-2319	9	3	many	many	ADJ
cana-2319	9	4	ways	way	NOUN
cana-2319	9	5	to	to	PART
cana-2319	9	6	compute	compute	VERB
cana-2319	9	7	the	the	DET
cana-2319	9	8	alexander	alexander	PROPN
cana-2319	9	9	polynomial	polynomial	NOUN
cana-2319	9	10	,	,	PUNCT
cana-2319	9	11	involving	involve	VERB
cana-2319	9	12	algebraic	algebraic	ADJ
cana-2319	9	13	techniques	technique	NOUN
cana-2319	9	14	and	and	CCONJ
cana-2319	9	15	others	other	NOUN
cana-2319	9	16	have	have	VERB
cana-2319	9	17	more	more	ADJ
cana-2319	9	18	geometric	geometric	ADJ
cana-2319	9	19	approaches	approach	NOUN
cana-2319	9	20	such	such	ADJ
cana-2319	9	21	as	as	ADP
cana-2319	9	22	[	[	X
cana-2319	9	23	1	1	NUM
cana-2319	9	24	]	]	PUNCT
cana-2319	9	25	,	,	PUNCT
cana-2319	9	26	[	[	X
cana-2319	9	27	2	2	NUM
cana-2319	9	28	]	]	PUNCT
cana-2319	9	29	,	,	PUNCT
cana-2319	9	30	[	[	X
cana-2319	9	31	4	4	NUM
cana-2319	9	32	]	]	PUNCT
cana-2319	9	33	,	,	PUNCT
cana-2319	9	34	[	[	X
cana-2319	9	35	6	6	NUM
cana-2319	9	36	]	]	PUNCT
cana-2319	9	37	,	,	PUNCT
cana-2319	9	38	...	...	PUNCT
cana-2319	9	39	in	in	ADP
cana-2319	9	40	this	this	DET
cana-2319	9	41	paper	paper	NOUN
cana-2319	9	42	,	,	PUNCT
cana-2319	9	43	we	we	PRON
cana-2319	9	44	will	will	AUX
cana-2319	9	45	see	see	VERB
cana-2319	9	46	connections	connection	NOUN
cana-2319	9	47	between	between	ADP
cana-2319	9	48	knot	knot	NOUN
cana-2319	9	49	theory	theory	NOUN
cana-2319	9	50	and	and	CCONJ
cana-2319	9	51	linear	linear	PROPN
cana-2319	9	52	algebra	algebra	NOUN
cana-2319	9	53	regarding	regard	VERB
cana-2319	9	54	two	two	NUM
cana-2319	9	55	knot	knot	ADJ
cana-2319	9	56	invariants	invariant	NOUN
cana-2319	9	57	,	,	PUNCT
cana-2319	9	58	the	the	DET
cana-2319	9	59	alexander	alexander	PROPN
cana-2319	9	60	polynomial	polynomial	PROPN
cana-2319	9	61	and	and	CCONJ
cana-2319	9	62	the	the	DET
cana-2319	9	63	3	3	NUM
cana-2319	9	64	-	-	PUNCT
cana-2319	9	65	coloring	coloring	NOUN
cana-2319	9	66	.	.	PUNCT
cana-2319	10	1	to	to	PART
cana-2319	10	2	prove	prove	VERB
cana-2319	10	3	that	that	SCONJ
cana-2319	10	4	the	the	DET
cana-2319	10	5	alexander	alexander	PROPN
cana-2319	10	6	polynomial	polynomial	PROPN
cana-2319	10	7	is	be	AUX
cana-2319	10	8	a	a	DET
cana-2319	10	9	knot	knot	NOUN
cana-2319	10	10	invariant	invariant	ADJ
cana-2319	10	11	is	be	AUX
cana-2319	10	12	not	not	PART
cana-2319	10	13	a	a	DET
cana-2319	10	14	new	new	ADJ
cana-2319	10	15	topic	topic	NOUN
cana-2319	10	16	,	,	PUNCT
cana-2319	10	17	it	it	PRON
cana-2319	10	18	is	be	AUX
cana-2319	10	19	already	already	ADV
cana-2319	10	20	proved	prove	VERB
cana-2319	10	21	in	in	ADP
cana-2319	10	22	several	several	ADJ
cana-2319	10	23	references	reference	NOUN
cana-2319	10	24	such	such	ADJ
cana-2319	10	25	as	as	ADP
cana-2319	10	26	[	[	X
cana-2319	10	27	1	1	NUM
cana-2319	10	28	]	]	PUNCT
cana-2319	10	29	,	,	PUNCT
cana-2319	10	30	[	[	X
cana-2319	10	31	2	2	NUM
cana-2319	10	32	]	]	PUNCT
cana-2319	10	33	,	,	PUNCT
cana-2319	10	34	[	[	X
cana-2319	10	35	3	3	NUM
cana-2319	10	36	]	]	PUNCT
cana-2319	10	37	,	,	PUNCT
cana-2319	10	38	[	[	X
cana-2319	10	39	4	4	NUM
cana-2319	10	40	]	]	PUNCT
cana-2319	10	41	,	,	PUNCT
cana-2319	10	42	[	[	X
cana-2319	10	43	7	7	NUM
cana-2319	10	44	]	]	PUNCT
cana-2319	10	45	.	.	PUNCT
cana-2319	11	1	we	we	PRON
cana-2319	11	2	claim	claim	VERB
cana-2319	11	3	that	that	SCONJ
cana-2319	11	4	we	we	PRON
cana-2319	11	5	touch	touch	VERB
cana-2319	11	6	a	a	DET
cana-2319	11	7	different	different	ADJ
cana-2319	11	8	approach	approach	NOUN
cana-2319	11	9	to	to	PART
cana-2319	11	10	prove	prove	VERB
cana-2319	11	11	such	such	DET
cana-2319	11	12	a	a	DET
cana-2319	11	13	well	well	ADV
cana-2319	11	14	-	-	PUNCT
cana-2319	11	15	known	know	VERB
cana-2319	11	16	property	property	NOUN
cana-2319	11	17	of	of	ADP
cana-2319	11	18	the	the	DET
cana-2319	11	19	alexander	alexander	PROPN
cana-2319	11	20	polynomial	polynomial	PROPN
cana-2319	11	21	and	and	CCONJ
cana-2319	11	22	suggest	suggest	VERB
cana-2319	11	23	a	a	DET
cana-2319	11	24	similar	similar	ADJ
cana-2319	11	25	proof	proof	NOUN
cana-2319	11	26	for	for	ADP
cana-2319	11	27	the	the	DET
cana-2319	11	28	same	same	ADJ
cana-2319	11	29	property	property	NOUN
cana-2319	11	30	of	of	ADP
cana-2319	11	31	the	the	DET
cana-2319	11	32	3coloring	3coloring	NUM
cana-2319	11	33	.	.	PUNCT
cana-2319	12	1	also	also	ADV
cana-2319	12	2	,	,	PUNCT
cana-2319	12	3	we	we	PRON
cana-2319	12	4	treated	treat	VERB
cana-2319	12	5	the	the	DET
cana-2319	12	6	3	3	NUM
cana-2319	12	7	-	-	PUNCT
cana-2319	12	8	coloring	color	VERB
cana-2319	12	9	invariant	invariant	NOUN
cana-2319	12	10	from	from	ADP
cana-2319	12	11	the	the	DET
cana-2319	12	12	linear	linear	PROPN
cana-2319	12	13	algebra	algebra	NOUN
cana-2319	12	14	point	point	NOUN
cana-2319	12	15	of	of	ADP
cana-2319	12	16	view	view	NOUN
cana-2319	12	17	depending	depend	VERB
cana-2319	12	18	on	on	ADP
cana-2319	12	19	the	the	DET
cana-2319	12	20	fox	fox	NOUN
cana-2319	12	21	coloring	coloring	NOUN
cana-2319	12	22	.	.	PUNCT
cana-2319	13	1	we	we	PRON
cana-2319	13	2	organized	organize	VERB
cana-2319	13	3	this	this	DET
cana-2319	13	4	paper	paper	NOUN
cana-2319	13	5	as	as	SCONJ
cana-2319	13	6	follows	follow	VERB
cana-2319	13	7	:	:	PUNCT
cana-2319	13	8	in	in	ADP
cana-2319	13	9	section	section	NOUN
cana-2319	13	10	2	2	NUM
cana-2319	13	11	,	,	PUNCT
cana-2319	13	12	we	we	PRON
cana-2319	13	13	give	give	VERB
cana-2319	13	14	a	a	DET
cana-2319	13	15	linear	linear	ADJ
cana-2319	13	16	algebra	algebra	NOUN
cana-2319	13	17	approach	approach	NOUN
cana-2319	13	18	of	of	ADP
cana-2319	13	19	computing	compute	VERB
cana-2319	13	20	the	the	DET
cana-2319	13	21	alexander	alexander	NOUN
cana-2319	13	22	polynomial	polynomial	NOUN
cana-2319	13	23	of	of	ADP
cana-2319	13	24	a	a	DET
cana-2319	13	25	knot	knot	NOUN
cana-2319	13	26	and	and	CCONJ
cana-2319	13	27	apply	apply	VERB
cana-2319	13	28	it	it	PRON
cana-2319	13	29	on	on	ADP
cana-2319	13	30	the	the	DET
cana-2319	13	31	trefoil	trefoil	NOUN
cana-2319	13	32	knot	knot	NOUN
cana-2319	13	33	.	.	PUNCT
cana-2319	14	1	in	in	ADP
cana-2319	14	2	section	section	NOUN
cana-2319	14	3	3	3	NUM
cana-2319	14	4	,	,	PUNCT
cana-2319	14	5	we	we	PRON
cana-2319	14	6	prepare	prepare	VERB
cana-2319	14	7	the	the	DET
cana-2319	14	8	reader	reader	NOUN
cana-2319	14	9	to	to	PART
cana-2319	14	10	accept	accept	VERB
cana-2319	14	11	that	that	PRON
cana-2319	14	12	playing	play	VERB
cana-2319	14	13	in	in	ADP
cana-2319	14	14	the	the	DET
cana-2319	14	15	given	give	VERB
cana-2319	14	16	oriented	orient	VERB
cana-2319	14	17	diagram	diagram	NOUN
cana-2319	14	18	of	of	ADP
cana-2319	14	19	a	a	DET
cana-2319	14	20	knot	knot	NOUN
cana-2319	14	21	in	in	ADP
cana-2319	14	22	a	a	DET
cana-2319	14	23	specific	specific	ADJ
cana-2319	14	24	way	way	NOUN
cana-2319	14	25	may	may	AUX
cana-2319	14	26	change	change	VERB
cana-2319	14	27	the	the	DET
cana-2319	14	28	form	form	NOUN
cana-2319	14	29	of	of	ADP
cana-2319	14	30	the	the	DET
cana-2319	14	31	corresponding	corresponding	ADJ
cana-2319	14	32	homogeneous	homogeneous	ADJ
cana-2319	14	33	system	system	NOUN
cana-2319	14	34	of	of	ADP
cana-2319	14	35	equations	equation	NOUN
cana-2319	14	36	that	that	PRON
cana-2319	14	37	can	can	AUX
cana-2319	14	38	be	be	AUX
cana-2319	14	39	obtained	obtain	VERB
cana-2319	14	40	from	from	ADP
cana-2319	14	41	a	a	DET
cana-2319	14	42	knot	knot	ADJ
cana-2319	14	43	diagram	diagram	NOUN
cana-2319	14	44	but	but	CCONJ
cana-2319	14	45	does	do	AUX
cana-2319	14	46	not	not	PART
cana-2319	14	47	change	change	VERB
cana-2319	14	48	its	its	PRON
cana-2319	14	49	space	space	NOUN
cana-2319	14	50	of	of	ADP
cana-2319	14	51	solutions	solution	NOUN
cana-2319	14	52	and	and	CCONJ
cana-2319	14	53	then	then	ADV
cana-2319	14	54	the	the	DET
cana-2319	14	55	alexander	alexander	PROPN
cana-2319	14	56	polynomial	polynomial	NOUN
cana-2319	14	57	of	of	ADP
cana-2319	14	58	the	the	DET
cana-2319	14	59	given	give	VERB
cana-2319	14	60	knot	knot	NOUN
cana-2319	14	61	is	be	AUX
cana-2319	14	62	unchanged	unchanged	ADJ
cana-2319	14	63	.	.	PUNCT
cana-2319	15	1	in	in	ADP
cana-2319	15	2	section	section	NOUN
cana-2319	15	3	4	4	NUM
cana-2319	15	4	,	,	PUNCT
cana-2319	15	5	we	we	PRON
cana-2319	15	6	put	put	VERB
cana-2319	15	7	an	an	DET
cana-2319	15	8	approach	approach	NOUN
cana-2319	15	9	to	to	PART
cana-2319	15	10	prove	prove	VERB
cana-2319	15	11	that	that	SCONJ
cana-2319	15	12	the	the	DET
cana-2319	15	13	alexander	alexander	PROPN
cana-2319	15	14	polynomial	polynomial	PROPN
cana-2319	15	15	is	be	AUX
cana-2319	15	16	a	a	DET
cana-2319	15	17	knot	knot	NOUN
cana-2319	15	18	invariant	invariant	ADJ
cana-2319	15	19	.	.	PUNCT
cana-2319	16	1	in	in	ADP
cana-2319	16	2	section	section	NOUN
cana-2319	16	3	5	5	NUM
cana-2319	16	4	,	,	PUNCT
cana-2319	16	5	we	we	PRON
cana-2319	16	6	connect	connect	VERB
cana-2319	16	7	the	the	DET
cana-2319	16	8	previous	previous	ADJ
cana-2319	16	9	approach	approach	NOUN
cana-2319	16	10	to	to	ADP
cana-2319	16	11	the	the	DET
cana-2319	16	12	fox	fox	NOUN
cana-2319	16	13	coloring	coloring	NOUN
cana-2319	16	14	way	way	NOUN
cana-2319	16	15	to	to	PART
cana-2319	16	16	suggest	suggest	VERB
cana-2319	16	17	a	a	DET
cana-2319	16	18	proof	proof	NOUN
cana-2319	16	19	of	of	ADP
cana-2319	16	20	the	the	DET
cana-2319	16	21	assumption	assumption	NOUN
cana-2319	16	22	that	that	SCONJ
cana-2319	16	23	the	the	DET
cana-2319	16	24	3	3	NUM
cana-2319	16	25	-	-	PUNCT
cana-2319	16	26	coloring	coloring	NOUN
cana-2319	16	27	is	be	AUX
cana-2319	16	28	a	a	DET
cana-2319	16	29	knot	knot	NOUN
cana-2319	16	30	invariant	invariant	ADJ
cana-2319	16	31	and	and	CCONJ
cana-2319	16	32	give	give	VERB
cana-2319	16	33	a	a	DET
cana-2319	16	34	way	way	NOUN
cana-2319	16	35	to	to	PART
cana-2319	16	36	check	check	VERB
cana-2319	16	37	if	if	SCONJ
cana-2319	16	38	a	a	DET
cana-2319	16	39	given	give	VERB
cana-2319	16	40	knot	knot	NOUN
cana-2319	16	41	diagram	diagram	NOUN
cana-2319	16	42	is	be	AUX
cana-2319	16	43	3	3	NUM
cana-2319	16	44	-	-	PUNCT
cana-2319	16	45	coloring	coloring	NOUN
cana-2319	16	46	or	or	CCONJ
cana-2319	16	47	not	not	PART
cana-2319	16	48	.	.	PUNCT
cana-2319	17	1	then	then	ADV
cana-2319	17	2	we	we	PRON
cana-2319	17	3	apply	apply	VERB
cana-2319	17	4	this	this	DET
cana-2319	17	5	approach	approach	NOUN
cana-2319	17	6	on	on	ADP
cana-2319	17	7	the	the	DET
cana-2319	17	8	trefoil	trefoil	NOUN
cana-2319	17	9	and	and	CCONJ
cana-2319	17	10	the	the	DET
cana-2319	17	11	figure	figure	NOUN
cana-2319	17	12	eight	eight	NUM
cana-2319	17	13	knots	knot	NOUN
cana-2319	17	14	depending	depend	VERB
cana-2319	17	15	on	on	ADP
cana-2319	17	16	the	the	DET
cana-2319	17	17	system	system	NOUN
cana-2319	17	18	obtained	obtain	VERB
cana-2319	17	19	in	in	ADP
cana-2319	17	20	section	section	NOUN
cana-2319	17	21	2	2	NUM
cana-2319	17	22	.	.	NOUN
cana-2319	17	23	2	2	NUM
cana-2319	17	24	.	.	X
cana-2319	18	1	the	the	DET
cana-2319	18	2	alexander	alexander	PROPN
cana-2319	18	3	polynomial	polynomial	PROPN
cana-2319	18	4	given	give	VERB
cana-2319	18	5	an	an	DET
cana-2319	18	6	oriented	orient	VERB
cana-2319	18	7	diagram	diagram	NOUN
cana-2319	18	8	of	of	ADP
cana-2319	18	9	a	a	DET
cana-2319	18	10	knot	knot	NOUN
cana-2319	18	11	𝐾	𝐾	PROPN
cana-2319	18	12	of	of	ADP
cana-2319	18	13	𝑛	𝑛	PROPN
cana-2319	18	14	crossings	crossing	NOUN
cana-2319	18	15	.	.	PUNCT
cana-2319	19	1	we	we	PRON
cana-2319	19	2	assign	assign	VERB
cana-2319	19	3	the	the	DET
cana-2319	19	4	variables	variable	NOUN
cana-2319	19	5	:	:	PUNCT
cana-2319	19	6	𝑥1	𝑥1	NOUN
cana-2319	19	7	,	,	PUNCT
cana-2319	19	8	𝑥2	𝑥2	NOUN
cana-2319	19	9	,	,	PUNCT
cana-2319	19	10	…	…	PUNCT
cana-2319	19	11	,	,	PUNCT
cana-2319	19	12	𝑥𝑛	𝑥𝑛	VERB
cana-2319	19	13	to	to	ADP
cana-2319	19	14	its	its	PRON
cana-2319	19	15	arcs	arc	NOUN
cana-2319	19	16	.	.	PUNCT
cana-2319	20	1	at	at	ADP
cana-2319	20	2	any	any	DET
cana-2319	20	3	crossing	crossing	NOUN
cana-2319	20	4	,	,	PUNCT
cana-2319	20	5	we	we	PRON
cana-2319	20	6	see	see	VERB
cana-2319	20	7	three	three	NUM
cana-2319	20	8	arcs	arc	NOUN
cana-2319	20	9	in	in	ADP
cana-2319	20	10	the	the	DET
cana-2319	20	11	position	position	NOUN
cana-2319	20	12	as	as	ADP
cana-2319	20	13	in	in	ADP
cana-2319	20	14	figure	figure	NOUN
cana-2319	20	15	2.1	2.1	NUM
cana-2319	20	16	.	.	PUNCT
cana-2319	21	1	mailto:ayman.aboufattoum@ku.edu.kw	mailto:ayman.aboufattoum@ku.edu.kw	DET
cana-2319	21	2	communications	communication	NOUN
cana-2319	21	3	on	on	ADP
cana-2319	21	4	applied	apply	VERB
cana-2319	21	5	nonlinear	nonlinear	ADJ
cana-2319	21	6	analysis	analysis	NOUN
cana-2319	21	7	issn	issn	NOUN
cana-2319	21	8	:	:	PUNCT
cana-2319	21	9	1074	1074	NUM
cana-2319	21	10	-	-	PUNCT
cana-2319	21	11	133x	133x	NUM
cana-2319	21	12	vol	vol	NOUN
cana-2319	21	13	32	32	NUM
cana-2319	21	14	no	no	NOUN
cana-2319	21	15	.	.	PUNCT
cana-2319	22	1	1s	1s	NUM
cana-2319	22	2	(	(	PUNCT
cana-2319	22	3	2025	2025	NUM
cana-2319	22	4	)	)	PUNCT
cana-2319	22	5	553	553	NUM
cana-2319	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2319	22	7	for	for	ADP
cana-2319	22	8	such	such	ADJ
cana-2319	22	9	crossing	crossing	NOUN
cana-2319	22	10	,	,	PUNCT
cana-2319	22	11	we	we	PRON
cana-2319	22	12	write	write	VERB
cana-2319	22	13	the	the	DET
cana-2319	22	14	equation	equation	NOUN
cana-2319	22	15	:	:	PUNCT
cana-2319	22	16	𝑡𝑥𝑖	𝑡𝑥𝑖	NOUN
cana-2319	22	17	+	+	CCONJ
cana-2319	22	18	(	(	PUNCT
cana-2319	22	19	1−	1−	NUM
cana-2319	22	20	𝑡)𝑥𝑖	𝑡)𝑥𝑖	PROPN
cana-2319	22	21	=	=	SYM
cana-2319	22	22	𝑥𝑘	𝑥𝑘	NOUN
cana-2319	22	23	,	,	PUNCT
cana-2319	22	24	𝑡	𝑡	PROPN
cana-2319	22	25	∈	∈	PROPN
cana-2319	22	26	ℂ	ℂ	PROPN
cana-2319	22	27	,	,	PUNCT
cana-2319	22	28	then	then	ADV
cana-2319	22	29	we	we	PRON
cana-2319	22	30	construct	construct	VERB
cana-2319	22	31	the	the	DET
cana-2319	22	32	following	follow	VERB
cana-2319	22	33	homogeneous	homogeneous	ADJ
cana-2319	22	34	system	system	NOUN
cana-2319	22	35	of	of	ADP
cana-2319	22	36	𝑛	𝑛	DET
cana-2319	22	37	linear	linear	ADJ
cana-2319	22	38	equations	equation	NOUN
cana-2319	22	39	in	in	ADP
cana-2319	22	40	the	the	DET
cana-2319	22	41	variables	variable	NOUN
cana-2319	22	42	:	:	PUNCT
cana-2319	22	43	𝑥1	𝑥1	NOUN
cana-2319	22	44	,	,	PUNCT
cana-2319	22	45	𝑥2	𝑥2	NOUN
cana-2319	22	46	,	,	PUNCT
cana-2319	22	47	…	…	PUNCT
cana-2319	22	48	,	,	PUNCT
cana-2319	22	49	𝑥𝑛	𝑥𝑛	PROPN
cana-2319	22	50	.	.	PUNCT
cana-2319	23	1	𝑡𝑥𝑖	𝑡𝑥𝑖	PROPN
cana-2319	24	1	+	+	CCONJ
cana-2319	24	2	(	(	PUNCT
cana-2319	24	3	1−	1−	NUM
cana-2319	24	4	𝑡)𝑥𝑖	𝑡)𝑥𝑖	PROPN
cana-2319	24	5	=	=	SYM
cana-2319	24	6	𝑥𝑘	𝑥𝑘	PROPN
cana-2319	24	7	,	,	PUNCT
cana-2319	24	8	𝑖	𝑖	PROPN
cana-2319	24	9	,	,	PUNCT
cana-2319	24	10	𝑘	𝑘	X
cana-2319	24	11	=	=	SYM
cana-2319	24	12	1	1	NUM
cana-2319	24	13	,	,	PUNCT
cana-2319	24	14	𝑛	𝑛	PRON
cana-2319	24	15	,	,	PUNCT
cana-2319	24	16	𝑡	𝑡	PROPN
cana-2319	24	17	∈	∈	PROPN
cana-2319	24	18	ℂ	ℂ	PROPN
cana-2319	24	19	(	(	PUNCT
cana-2319	24	20	2.1	2.1	NUM
cana-2319	24	21	)	)	PUNCT
cana-2319	24	22	this	this	DET
cana-2319	24	23	system	system	NOUN
cana-2319	24	24	can	can	AUX
cana-2319	24	25	be	be	AUX
cana-2319	24	26	also	also	ADV
cana-2319	24	27	written	write	VERB
cana-2319	24	28	in	in	ADP
cana-2319	24	29	the	the	DET
cana-2319	24	30	matrix	matrix	NOUN
cana-2319	24	31	form	form	NOUN
cana-2319	24	32	,	,	PUNCT
cana-2319	24	33	𝐴𝑋	𝐴𝑋	PROPN
cana-2319	24	34	=	=	SYM
cana-2319	24	35	0⃑	0⃑	NOUN
cana-2319	24	36	.	.	PUNCT
cana-2319	25	1	because	because	SCONJ
cana-2319	25	2	of	of	ADP
cana-2319	25	3	this	this	DET
cana-2319	25	4	way	way	NOUN
cana-2319	25	5	of	of	ADP
cana-2319	25	6	construction	construction	NOUN
cana-2319	25	7	,	,	PUNCT
cana-2319	25	8	the	the	DET
cana-2319	25	9	entries	entry	NOUN
cana-2319	25	10	of	of	ADP
cana-2319	25	11	any	any	DET
cana-2319	25	12	row	row	NOUN
cana-2319	25	13	of	of	ADP
cana-2319	25	14	𝐴	𝐴	PROPN
cana-2319	25	15	are	be	AUX
cana-2319	25	16	𝑡	𝑡	NOUN
cana-2319	25	17	,	,	PUNCT
cana-2319	25	18	1−	1−	NUM
cana-2319	25	19	𝑡,−1	𝑡,−1	NOUN
cana-2319	25	20	,	,	PUNCT
cana-2319	25	21	and	and	CCONJ
cana-2319	25	22	some	some	DET
cana-2319	25	23	possible	possible	ADJ
cana-2319	25	24	zeros	zero	NOUN
cana-2319	25	25	then	then	ADV
cana-2319	25	26	𝐴	𝐴	PROPN
cana-2319	25	27	is	be	AUX
cana-2319	25	28	singular	singular	ADJ
cana-2319	25	29	for	for	ADP
cana-2319	25	30	every	every	DET
cana-2319	25	31	𝑡.	𝑡.	NOUN
cana-2319	25	32	in	in	ADP
cana-2319	25	33	the	the	DET
cana-2319	25	34	1920s	1920s	NUM
cana-2319	25	35	,	,	PUNCT
cana-2319	25	36	alexander	alexander	PROPN
cana-2319	25	37	showed	show	VERB
cana-2319	25	38	that	that	SCONJ
cana-2319	25	39	all	all	DET
cana-2319	25	40	(	(	PUNCT
cana-2319	25	41	𝑛	𝑛	DET
cana-2319	25	42	−	−	PROPN
cana-2319	25	43	1)-minors	1)-minors	NUM
cana-2319	25	44	are	be	AUX
cana-2319	25	45	equal	equal	ADJ
cana-2319	25	46	up	up	ADP
cana-2319	25	47	to	to	ADP
cana-2319	25	48	multiplication	multiplication	NOUN
cana-2319	25	49	by	by	ADP
cana-2319	25	50	the	the	DET
cana-2319	25	51	monomial	monomial	ADJ
cana-2319	25	52	±𝑡𝑘	±𝑡𝑘	NOUN
cana-2319	25	53	,	,	PUNCT
cana-2319	25	54	𝑘	𝑘	PROPN
cana-2319	25	55	∈	∈	PROPN
cana-2319	25	56	ℤ.	ℤ.	PROPN
cana-2319	25	57	so	so	ADV
cana-2319	25	58	,	,	PUNCT
cana-2319	25	59	these	these	DET
cana-2319	25	60	minors	minor	NOUN
cana-2319	25	61	which	which	PRON
cana-2319	25	62	are	be	AUX
cana-2319	25	63	laurent	laurent	NOUN
cana-2319	25	64	polynomials	polynomial	NOUN
cana-2319	25	65	in	in	ADP
cana-2319	25	66	𝑡	𝑡	PROPN
cana-2319	25	67	can	can	AUX
cana-2319	25	68	be	be	AUX
cana-2319	25	69	normalized	normalize	VERB
cana-2319	25	70	for	for	ADP
cana-2319	25	71	some	some	DET
cana-2319	25	72	consideration	consideration	NOUN
cana-2319	25	73	by	by	ADP
cana-2319	25	74	dividing	divide	VERB
cana-2319	25	75	a	a	DET
cana-2319	25	76	minor	minor	NOUN
cana-2319	25	77	by	by	ADP
cana-2319	25	78	±𝑡𝑘.	±𝑡𝑘.	NOUN
cana-2319	25	79	any	any	DET
cana-2319	25	80	one	one	NUM
cana-2319	25	81	of	of	ADP
cana-2319	25	82	these	these	DET
cana-2319	25	83	polynomials	polynomial	NOUN
cana-2319	25	84	is	be	AUX
cana-2319	25	85	called	call	VERB
cana-2319	25	86	an	an	DET
cana-2319	25	87	alexander	alexander	NOUN
cana-2319	25	88	polynomial	polynomial	NOUN
cana-2319	25	89	of	of	ADP
cana-2319	25	90	the	the	DET
cana-2319	25	91	knot	knot	PROPN
cana-2319	25	92	𝐾	𝐾	PROPN
cana-2319	25	93	,	,	PUNCT
cana-2319	25	94	denoted	denote	VERB
cana-2319	25	95	by	by	ADP
cana-2319	25	96	∇𝐾(𝑡	∇𝐾(𝑡	NOUN
cana-2319	25	97	)	)	PUNCT
cana-2319	25	98	with	with	ADP
cana-2319	25	99	∇ο(𝑡	∇ο(𝑡	NOUN
cana-2319	25	100	)	)	PUNCT
cana-2319	25	101	=	=	SYM
cana-2319	26	1	1	1	X
cana-2319	26	2	.	.	X
cana-2319	26	3	one	one	NUM
cana-2319	26	4	often	often	ADV
cana-2319	26	5	fixes	fix	VERB
cana-2319	26	6	a	a	DET
cana-2319	26	7	particular	particular	ADJ
cana-2319	26	8	form	form	NOUN
cana-2319	26	9	as	as	ADP
cana-2319	26	10	the	the	DET
cana-2319	26	11	first	first	ADJ
cana-2319	26	12	alexander	alexander	NOUN
cana-2319	26	13	's	's	PART
cana-2319	26	14	choice	choice	NOUN
cana-2319	26	15	of	of	ADP
cana-2319	26	16	the	the	DET
cana-2319	26	17	normalization	normalization	NOUN
cana-2319	26	18	so	so	SCONJ
cana-2319	26	19	that	that	SCONJ
cana-2319	26	20	the	the	DET
cana-2319	26	21	term	term	NOUN
cana-2319	26	22	of	of	ADP
cana-2319	26	23	the	the	DET
cana-2319	26	24	polynomial	polynomial	NOUN
cana-2319	26	25	of	of	ADP
cana-2319	26	26	the	the	DET
cana-2319	26	27	least	least	ADJ
cana-2319	26	28	degree	degree	NOUN
cana-2319	26	29	of	of	ADP
cana-2319	26	30	𝑡	𝑡	PROPN
cana-2319	26	31	is	be	AUX
cana-2319	26	32	a	a	DET
cana-2319	26	33	positive	positive	ADJ
cana-2319	26	34	constant	constant	NOUN
cana-2319	26	35	.	.	PUNCT
cana-2319	27	1	a	a	DET
cana-2319	27	2	simple	simple	ADJ
cana-2319	27	3	example	example	NOUN
cana-2319	27	4	is	be	AUX
cana-2319	27	5	now	now	ADV
cana-2319	27	6	introduced	introduce	VERB
cana-2319	27	7	to	to	PART
cana-2319	27	8	visualize	visualize	VERB
cana-2319	27	9	the	the	DET
cana-2319	27	10	process	process	NOUN
cana-2319	27	11	of	of	ADP
cana-2319	27	12	computing	compute	VERB
cana-2319	27	13	the	the	DET
cana-2319	27	14	alexander	alexander	PROPN
cana-2319	27	15	polynomial	polynomial	PROPN
cana-2319	27	16	using	use	VERB
cana-2319	27	17	this	this	DET
cana-2319	27	18	approach	approach	NOUN
cana-2319	27	19	.	.	PUNCT
cana-2319	28	1	let	let	VERB
cana-2319	28	2	us	we	PRON
cana-2319	28	3	use	use	VERB
cana-2319	28	4	this	this	DET
cana-2319	28	5	approach	approach	NOUN
cana-2319	28	6	to	to	PART
cana-2319	28	7	compute	compute	VERB
cana-2319	28	8	the	the	DET
cana-2319	28	9	alexander	alexander	NOUN
cana-2319	28	10	polynomial	polynomial	NOUN
cana-2319	28	11	of	of	ADP
cana-2319	28	12	the	the	DET
cana-2319	28	13	trefoil	trefoil	NOUN
cana-2319	28	14	.	.	PUNCT
cana-2319	29	1	the	the	DET
cana-2319	29	2	corresponding	corresponding	ADJ
cana-2319	29	3	linear	linear	NOUN
cana-2319	29	4	system	system	NOUN
cana-2319	29	5	is	be	AUX
cana-2319	29	6	𝑡𝑧	𝑡𝑧	NOUN
cana-2319	29	7	+	+	X
cana-2319	29	8	(	(	PUNCT
cana-2319	29	9	1−	1−	NUM
cana-2319	29	10	𝑡)𝑥	𝑡)𝑥	NUM
cana-2319	29	11	=	=	PUNCT
cana-2319	29	12	𝑧	𝑧	PRON
cana-2319	29	13	𝑡𝑥	𝑡𝑥	NOUN
cana-2319	29	14	+	+	CCONJ
cana-2319	29	15	(	(	PUNCT
cana-2319	29	16	1−	1−	NUM
cana-2319	29	17	𝑡)𝑦	𝑡)𝑦	NOUN
cana-2319	29	18	=	=	PUNCT
cana-2319	29	19	𝑧	𝑧	NOUN
cana-2319	29	20	𝑡𝑦	𝑡𝑦	NOUN
cana-2319	29	21	+	+	CCONJ
cana-2319	29	22	(	(	PUNCT
cana-2319	29	23	1−	1−	NUM
cana-2319	29	24	𝑡)𝑧	𝑡)𝑧	NOUN
cana-2319	29	25	=	=	SYM
cana-2319	29	26	𝑥	𝑥	X
cana-2319	29	27	which	which	PRON
cana-2319	29	28	can	can	AUX
cana-2319	29	29	be	be	AUX
cana-2319	29	30	rewritten	rewrite	VERB
cana-2319	29	31	as	as	ADP
cana-2319	29	32	:	:	PUNCT
cana-2319	29	33	[	[	PUNCT
cana-2319	29	34	1−	1−	NUM
cana-2319	29	35	𝑡	𝑡	NUM
cana-2319	29	36	−1	−1	NOUN
cana-2319	29	37	𝑡	𝑡	PROPN
cana-2319	29	38	𝑡	𝑡	PROPN
cana-2319	29	39	1−	1−	NUM
cana-2319	29	40	𝑡	𝑡	SYM
cana-2319	29	41	−1	−1	NOUN
cana-2319	29	42	−1	−1	NOUN
cana-2319	29	43	𝑡	𝑡	PROPN
cana-2319	29	44	1−	1−	NUM
cana-2319	29	45	𝑡	𝑡	X
cana-2319	29	46	]	]	PUNCT
cana-2319	29	47	[	[	PUNCT
cana-2319	29	48	𝑥	𝑥	X
cana-2319	29	49	𝑦	𝑦	NOUN
cana-2319	29	50	𝑧	𝑧	X
cana-2319	29	51	]	]	PUNCT
cana-2319	29	52	=	=	PUNCT
cana-2319	30	1	[	[	PUNCT
cana-2319	30	2	0	0	NUM
cana-2319	30	3	0	0	NUM
cana-2319	30	4	0	0	NUM
cana-2319	30	5	]	]	PUNCT
cana-2319	30	6	take	take	VERB
cana-2319	30	7	any	any	DET
cana-2319	30	8	2−minor	2−minor	NOUN
cana-2319	30	9	,	,	PUNCT
cana-2319	30	10	say	say	VERB
cana-2319	30	11	|	|	ADV
cana-2319	30	12	𝑡	𝑡	NOUN
cana-2319	30	13	−1	−1	NOUN
cana-2319	30	14	−1	−1	NOUN
cana-2319	30	15	1−	1−	NUM
cana-2319	30	16	𝑡	𝑡	NOUN
cana-2319	30	17	|	|	NOUN
cana-2319	30	18	=	=	SYM
cana-2319	30	19	𝑡	𝑡	PROPN
cana-2319	30	20	−	−	NOUN
cana-2319	30	21	𝑡2	𝑡2	NOUN
cana-2319	30	22	+	+	CCONJ
cana-2319	30	23	1	1	NUM
cana-2319	30	24	,	,	PUNCT
cana-2319	30	25	or	or	CCONJ
cana-2319	30	26	alternatively	alternatively	ADV
cana-2319	30	27	,	,	PUNCT
cana-2319	30	28	if	if	SCONJ
cana-2319	30	29	we	we	PRON
cana-2319	30	30	take	take	VERB
cana-2319	30	31	|	|	ADV
cana-2319	30	32	𝑡	𝑡	NOUN
cana-2319	30	33	−1	−1	NOUN
cana-2319	30	34	−1	−1	NOUN
cana-2319	30	35	1−	1−	NUM
cana-2319	31	1	𝑡	𝑡	NOUN
cana-2319	31	2	|	|	NOUN
cana-2319	31	3	=	=	SYM
cana-2319	31	4	𝑡	𝑡	PROPN
cana-2319	31	5	−	−	NOUN
cana-2319	31	6	𝑡2	𝑡2	NOUN
cana-2319	31	7	−	−	PROPN
cana-2319	31	8	1	1	NUM
cana-2319	31	9	,	,	PUNCT
cana-2319	31	10	which	which	PRON
cana-2319	31	11	can	can	AUX
cana-2319	31	12	be	be	AUX
cana-2319	31	13	normalized	normalize	VERB
cana-2319	31	14	to	to	ADP
cana-2319	31	15	the	the	DET
cana-2319	31	16	previous	previous	ADJ
cana-2319	31	17	one	one	NUM
cana-2319	31	18	.	.	PUNCT
cana-2319	32	1	3	3	X
cana-2319	32	2	.	.	X
cana-2319	32	3	a	a	DET
cana-2319	32	4	useful	useful	ADJ
cana-2319	32	5	definition	definition	NOUN
cana-2319	32	6	we	we	PRON
cana-2319	32	7	claim	claim	VERB
cana-2319	32	8	that	that	SCONJ
cana-2319	32	9	getting	get	VERB
cana-2319	32	10	a	a	DET
cana-2319	32	11	homogeneous	homogeneous	ADJ
cana-2319	32	12	linear	linear	NOUN
cana-2319	32	13	system	system	NOUN
cana-2319	32	14	of	of	ADP
cana-2319	32	15	a	a	DET
cana-2319	32	16	knot	knot	NOUN
cana-2319	32	17	and	and	CCONJ
cana-2319	32	18	preserving	preserve	VERB
cana-2319	32	19	its	its	PRON
cana-2319	32	20	solutions	solution	NOUN
cana-2319	32	21	from	from	ADP
cana-2319	32	22	the	the	DET
cana-2319	32	23	change	change	NOUN
cana-2319	32	24	guarantees	guarantee	VERB
cana-2319	32	25	that	that	SCONJ
cana-2319	32	26	the	the	DET
cana-2319	32	27	alexander	alexander	PROPN
cana-2319	32	28	polynomial	polynomial	NOUN
cana-2319	32	29	of	of	ADP
cana-2319	32	30	the	the	DET
cana-2319	32	31	same	same	ADJ
cana-2319	32	32	knot	knot	NOUN
cana-2319	32	33	will	will	AUX
cana-2319	32	34	not	not	PART
cana-2319	32	35	change	change	VERB
cana-2319	32	36	up	up	ADP
cana-2319	32	37	to	to	ADP
cana-2319	32	38	multiplication	multiplication	NOUN
cana-2319	32	39	of	of	ADP
cana-2319	32	40	±𝑡𝑘	±𝑡𝑘	NOUN
cana-2319	32	41	,	,	PUNCT
cana-2319	32	42	for	for	ADP
cana-2319	32	43	some	some	DET
cana-2319	32	44	integer	integer	NOUN
cana-2319	32	45	𝑘.	𝑘.	NOUN
cana-2319	32	46	a	a	DET
cana-2319	32	47	very	very	ADV
cana-2319	32	48	close	close	ADJ
cana-2319	32	49	claim	claim	NOUN
cana-2319	32	50	is	be	AUX
cana-2319	32	51	mentioned	mention	VERB
cana-2319	32	52	and	and	CCONJ
cana-2319	32	53	proved	prove	VERB
cana-2319	32	54	in	in	ADP
cana-2319	32	55	[	[	X
cana-2319	32	56	2	2	NUM
cana-2319	32	57	]	]	PUNCT
cana-2319	32	58	.	.	PUNCT
cana-2319	33	1	as	as	ADP
cana-2319	33	2	in	in	ADP
cana-2319	33	3	the	the	DET
cana-2319	33	4	basics	basic	NOUN
cana-2319	33	5	of	of	ADP
cana-2319	33	6	linear	linear	PROPN
cana-2319	33	7	algebra	algebra	NOUN
cana-2319	33	8	,	,	PUNCT
cana-2319	33	9	where	where	SCONJ
cana-2319	33	10	we	we	PRON
cana-2319	33	11	can	can	AUX
cana-2319	33	12	check	check	VERB
cana-2319	33	13	if	if	SCONJ
cana-2319	33	14	two	two	NUM
cana-2319	33	15	given	give	VERB
cana-2319	33	16	matrices	matrix	NOUN
cana-2319	33	17	of	of	ADP
cana-2319	33	18	the	the	DET
cana-2319	33	19	same	same	ADJ
cana-2319	33	20	size	size	NOUN
cana-2319	33	21	are	be	AUX
cana-2319	33	22	row	row	NOUN
cana-2319	33	23	equivalent	equivalent	ADJ
cana-2319	33	24	or	or	CCONJ
cana-2319	33	25	not	not	PART
cana-2319	33	26	,	,	PUNCT
cana-2319	33	27	communications	communication	NOUN
cana-2319	33	28	on	on	ADP
cana-2319	33	29	applied	apply	VERB
cana-2319	33	30	nonlinear	nonlinear	ADJ
cana-2319	33	31	analysis	analysis	NOUN
cana-2319	33	32	issn	issn	NOUN
cana-2319	33	33	:	:	PUNCT
cana-2319	33	34	1074	1074	NUM
cana-2319	33	35	-	-	PUNCT
cana-2319	33	36	133x	133x	NUM
cana-2319	33	37	vol	vol	NOUN
cana-2319	33	38	32	32	NUM
cana-2319	33	39	no	no	NOUN
cana-2319	33	40	.	.	PUNCT
cana-2319	34	1	1s	1s	NUM
cana-2319	34	2	(	(	PUNCT
cana-2319	34	3	2025	2025	NUM
cana-2319	34	4	)	)	PUNCT
cana-2319	34	5	554	554	NUM
cana-2319	34	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2319	35	1	the	the	DET
cana-2319	35	2	following	follow	VERB
cana-2319	35	3	definition	definition	NOUN
cana-2319	35	4	from	from	ADP
cana-2319	35	5	[	[	X
cana-2319	35	6	7	7	NUM
cana-2319	35	7	]	]	PUNCT
cana-2319	35	8	helps	help	VERB
cana-2319	35	9	us	we	PRON
cana-2319	35	10	to	to	PART
cana-2319	35	11	extend	extend	VERB
cana-2319	35	12	the	the	DET
cana-2319	35	13	concept	concept	NOUN
cana-2319	35	14	of	of	ADP
cana-2319	35	15	equivalency	equivalency	NOUN
cana-2319	35	16	regarding	regard	VERB
cana-2319	35	17	the	the	DET
cana-2319	35	18	solutions	solution	NOUN
cana-2319	35	19	of	of	ADP
cana-2319	35	20	the	the	DET
cana-2319	35	21	corresponding	corresponding	ADJ
cana-2319	35	22	homogeneous	homogeneous	ADJ
cana-2319	35	23	linear	linear	PROPN
cana-2319	35	24	systems	system	NOUN
cana-2319	35	25	of	of	ADP
cana-2319	35	26	equations	equation	NOUN
cana-2319	35	27	.	.	PUNCT
cana-2319	36	1	definition	definition	NOUN
cana-2319	36	2	.	.	PUNCT
cana-2319	37	1	two	two	NUM
cana-2319	37	2	matrices	matrix	NOUN
cana-2319	37	3	are	be	AUX
cana-2319	37	4	equivalent	equivalent	ADJ
cana-2319	37	5	if	if	SCONJ
cana-2319	37	6	one	one	PRON
cana-2319	37	7	can	can	AUX
cana-2319	37	8	be	be	AUX
cana-2319	37	9	obtained	obtain	VERB
cana-2319	37	10	from	from	ADP
cana-2319	37	11	the	the	DET
cana-2319	37	12	other	other	ADJ
cana-2319	37	13	by	by	ADP
cana-2319	37	14	a	a	DET
cana-2319	37	15	finite	finite	ADJ
cana-2319	37	16	sequence	sequence	NOUN
cana-2319	37	17	of	of	ADP
cana-2319	37	18	the	the	DET
cana-2319	37	19	following	follow	VERB
cana-2319	37	20	operations	operation	NOUN
cana-2319	37	21	and	and	CCONJ
cana-2319	37	22	their	their	PRON
cana-2319	37	23	inverse	inverse	NOUN
cana-2319	37	24	operations	operation	NOUN
cana-2319	37	25	:	:	PUNCT
cana-2319	38	1	1	1	X
cana-2319	38	2	.	.	X
cana-2319	38	3	permute	permute	PROPN
cana-2319	38	4	two	two	NUM
cana-2319	38	5	rows(columns	rows(column	NOUN
cana-2319	38	6	)	)	PUNCT
cana-2319	38	7	.	.	PUNCT
cana-2319	39	1	2	2	X
cana-2319	39	2	.	.	X
cana-2319	39	3	adjoin	adjoin	VERB
cana-2319	39	4	a	a	DET
cana-2319	39	5	new	new	ADJ
cana-2319	39	6	row	row	NOUN
cana-2319	39	7	of	of	ADP
cana-2319	39	8	zeros	zero	NOUN
cana-2319	39	9	.	.	PUNCT
cana-2319	40	1	3	3	X
cana-2319	40	2	.	.	X
cana-2319	40	3	add	add	VERB
cana-2319	40	4	a	a	DET
cana-2319	40	5	multiple	multiple	NOUN
cana-2319	40	6	of	of	ADP
cana-2319	40	7	a	a	DET
cana-2319	40	8	row(column	row(column	NOUN
cana-2319	40	9	)	)	PUNCT
cana-2319	40	10	to	to	ADP
cana-2319	40	11	any	any	DET
cana-2319	40	12	row(column	row(column	NOUN
cana-2319	40	13	)	)	PUNCT
cana-2319	40	14	.	.	PUNCT
cana-2319	41	1	4	4	X
cana-2319	41	2	.	.	X
cana-2319	41	3	multiply	multiply	VERB
cana-2319	41	4	a	a	DET
cana-2319	41	5	row(column	row(column	NOUN
cana-2319	41	6	)	)	PUNCT
cana-2319	41	7	by	by	ADP
cana-2319	41	8	a	a	DET
cana-2319	41	9	unit	unit	NOUN
cana-2319	41	10	.	.	PUNCT
cana-2319	42	1	5	5	X
cana-2319	42	2	.	.	X
cana-2319	42	3	replace	replace	VERB
cana-2319	42	4	the	the	DET
cana-2319	42	5	𝑚	𝑚	PROPN
cana-2319	42	6	×	×	NOUN
cana-2319	42	7	𝑛	𝑛	PRON
cana-2319	42	8	matrix	matrix	NOUN
cana-2319	42	9	𝐴	𝐴	PROPN
cana-2319	42	10	by	by	ADP
cana-2319	42	11	the	the	DET
cana-2319	42	12	extended	extended	ADJ
cana-2319	42	13	(	(	PUNCT
cana-2319	42	14	𝑚	𝑚	X
cana-2319	42	15	+	+	NOUN
cana-2319	42	16	1	1	NUM
cana-2319	42	17	)	)	PUNCT
cana-2319	42	18	×	×	NOUN
cana-2319	42	19	(	(	PUNCT
cana-2319	42	20	𝑛	𝑛	PROPN
cana-2319	42	21	+	+	NOUN
cana-2319	42	22	1)-matrix	1)-matrix	NUM
cana-2319	42	23	:	:	PUNCT
cana-2319	42	24	[	[	PUNCT
cana-2319	42	25	𝐴	𝐴	NOUN
cana-2319	42	26	0	0	NUM
cana-2319	42	27	⋮	⋮	NOUN
cana-2319	42	28	0	0	NUM
cana-2319	42	29	…	…	PUNCT
cana-2319	42	30	0	0	NUM
cana-2319	42	31	0	0	NUM
cana-2319	42	32	1	1	NUM
cana-2319	42	33	]	]	SYM
cana-2319	42	34	4	4	X
cana-2319	42	35	.	.	X
cana-2319	43	1	a	a	DET
cana-2319	43	2	linear	linear	ADJ
cana-2319	43	3	algebra	algebra	NOUN
cana-2319	43	4	approach	approach	NOUN
cana-2319	43	5	to	to	ADP
cana-2319	43	6	the	the	DET
cana-2319	43	7	proof	proof	NOUN
cana-2319	43	8	of	of	ADP
cana-2319	43	9	an	an	DET
cana-2319	43	10	old	old	ADJ
cana-2319	43	11	theorem	theorem	NOUN
cana-2319	43	12	theorem	theorem	NOUN
cana-2319	43	13	.	.	PROPN
cana-2319	43	14	∇𝐾(𝑡	∇𝐾(𝑡	X
cana-2319	43	15	)	)	PUNCT
cana-2319	43	16	is	be	AUX
cana-2319	43	17	a	a	DET
cana-2319	43	18	knot	knot	NOUN
cana-2319	43	19	invariant	invariant	ADJ
cana-2319	43	20	.	.	PUNCT
cana-2319	44	1	proof	proof	NOUN
cana-2319	44	2	.	.	PUNCT
cana-2319	45	1	to	to	PART
cana-2319	45	2	show	show	VERB
cana-2319	45	3	that	that	SCONJ
cana-2319	45	4	∇𝐾(𝑡	∇𝐾(𝑡	NOUN
cana-2319	45	5	)	)	PUNCT
cana-2319	45	6	is	be	AUX
cana-2319	45	7	a	a	DET
cana-2319	45	8	knot	knot	NOUN
cana-2319	45	9	invariant	invariant	ADJ
cana-2319	45	10	it	it	PRON
cana-2319	45	11	is	be	AUX
cana-2319	45	12	sufficient	sufficient	ADJ
cana-2319	45	13	to	to	PART
cana-2319	45	14	show	show	VERB
cana-2319	45	15	that	that	SCONJ
cana-2319	45	16	it	it	PRON
cana-2319	45	17	does	do	AUX
cana-2319	45	18	not	not	PART
cana-2319	45	19	change	change	VERB
cana-2319	45	20	under	under	ADP
cana-2319	45	21	reidemeister	reidemeister	ADJ
cana-2319	45	22	moves	move	NOUN
cana-2319	45	23	.	.	PUNCT
cana-2319	46	1	for	for	ADP
cana-2319	46	2	the	the	DET
cana-2319	46	3	first	first	ADJ
cana-2319	46	4	reidemeister	reidemeister	ADJ
cana-2319	46	5	move	move	NOUN
cana-2319	46	6	,	,	PUNCT
cana-2319	46	7	a	a	DET
cana-2319	46	8	twist	twist	NOUN
cana-2319	46	9	is	be	AUX
cana-2319	46	10	introduced	introduce	VERB
cana-2319	46	11	to	to	ADP
cana-2319	46	12	any	any	DET
cana-2319	46	13	arc	arc	NOUN
cana-2319	46	14	of	of	ADP
cana-2319	46	15	the	the	DET
cana-2319	46	16	given	give	VERB
cana-2319	46	17	oriented	orient	VERB
cana-2319	46	18	knot	knot	NOUN
cana-2319	46	19	diagram	diagram	NOUN
cana-2319	46	20	,	,	PUNCT
cana-2319	46	21	say	say	INTJ
cana-2319	46	22	,	,	PUNCT
cana-2319	46	23	𝑥𝑖	𝑥𝑖	PROPN
cana-2319	46	24	as	as	ADP
cana-2319	46	25	in	in	ADP
cana-2319	46	26	figure	figure	NOUN
cana-2319	46	27	4.1	4.1	NUM
cana-2319	46	28	.	.	PUNCT
cana-2319	47	1	keeping	keep	VERB
cana-2319	47	2	the	the	DET
cana-2319	47	3	whole	whole	ADJ
cana-2319	47	4	diagram	diagram	NOUN
cana-2319	47	5	fixed	fix	VERB
cana-2319	47	6	except	except	SCONJ
cana-2319	47	7	this	this	DET
cana-2319	47	8	move	move	NOUN
cana-2319	47	9	at	at	ADP
cana-2319	47	10	the	the	DET
cana-2319	47	11	arc	arc	NOUN
cana-2319	47	12	𝑥𝑖.	𝑥𝑖.	NOUN
cana-2319	47	13	to	to	PART
cana-2319	47	14	see	see	VERB
cana-2319	47	15	the	the	DET
cana-2319	47	16	contribution	contribution	NOUN
cana-2319	47	17	of	of	ADP
cana-2319	47	18	this	this	DET
cana-2319	47	19	move	move	NOUN
cana-2319	47	20	to	to	ADP
cana-2319	47	21	the	the	DET
cana-2319	47	22	system	system	NOUN
cana-2319	47	23	2.1	2.1	NUM
cana-2319	47	24	of	of	ADP
cana-2319	47	25	the	the	DET
cana-2319	47	26	given	give	VERB
cana-2319	47	27	diagram	diagram	NOUN
cana-2319	47	28	,	,	PUNCT
cana-2319	47	29	the	the	DET
cana-2319	47	30	new	new	ADJ
cana-2319	47	31	twist	twist	NOUN
cana-2319	47	32	adds	add	VERB
cana-2319	47	33	the	the	DET
cana-2319	47	34	equation	equation	NOUN
cana-2319	47	35	:	:	PUNCT
cana-2319	47	36	(	(	PUNCT
cana-2319	47	37	1−	1−	X
cana-2319	47	38	𝑡)𝑥𝑖	𝑡)𝑥𝑖	PROPN
cana-2319	47	39	+	+	PUNCT
cana-2319	47	40	𝑡𝑥𝑖	𝑡𝑥𝑖	NOUN
cana-2319	47	41	=	=	SYM
cana-2319	47	42	𝑥𝑖	𝑥𝑖	PROPN
cana-2319	47	43	,	,	PUNCT
cana-2319	47	44	which	which	PRON
cana-2319	47	45	does	do	AUX
cana-2319	47	46	not	not	PART
cana-2319	47	47	affect	affect	VERB
cana-2319	47	48	the	the	DET
cana-2319	47	49	solutions	solution	NOUN
cana-2319	47	50	of	of	ADP
cana-2319	47	51	the	the	DET
cana-2319	47	52	system	system	NOUN
cana-2319	47	53	.	.	PUNCT
cana-2319	48	1	it	it	PRON
cana-2319	48	2	is	be	AUX
cana-2319	48	3	typically	typically	ADV
cana-2319	48	4	,	,	PUNCT
cana-2319	48	5	as	as	SCONJ
cana-2319	48	6	one	one	NUM
cana-2319	48	7	adjoined	adjoin	VERB
cana-2319	48	8	a	a	DET
cana-2319	48	9	row	row	NOUN
cana-2319	48	10	of	of	ADP
cana-2319	48	11	zeros	zero	NOUN
cana-2319	48	12	to	to	ADP
cana-2319	48	13	the	the	DET
cana-2319	48	14	coefficients	coefficient	NOUN
cana-2319	48	15	matrix	matrix	NOUN
cana-2319	48	16	of	of	ADP
cana-2319	48	17	the	the	DET
cana-2319	48	18	system	system	NOUN
cana-2319	48	19	𝐴𝑋	𝐴𝑋	PROPN
cana-2319	48	20	=	=	SYM
cana-2319	48	21	0⃑	0⃑	NOUN
cana-2319	48	22	.	.	PUNCT
cana-2319	49	1	then	then	ADV
cana-2319	49	2	∇𝐾(𝑡	∇𝐾(𝑡	ADV
cana-2319	49	3	)	)	PUNCT
cana-2319	49	4	is	be	AUX
cana-2319	49	5	unchanged	unchanged	ADJ
cana-2319	49	6	under	under	ADP
cana-2319	49	7	the	the	DET
cana-2319	49	8	first	first	ADJ
cana-2319	49	9	reidemeister	reidemeister	ADJ
cana-2319	49	10	move	move	NOUN
cana-2319	49	11	.	.	PUNCT
cana-2319	50	1	note	note	NOUN
cana-2319	50	2	.	.	PUNCT
cana-2319	51	1	the	the	DET
cana-2319	51	2	other	other	ADJ
cana-2319	51	3	type	type	NOUN
cana-2319	51	4	of	of	ADP
cana-2319	51	5	the	the	DET
cana-2319	51	6	twist	twist	NOUN
cana-2319	51	7	will	will	AUX
cana-2319	51	8	add	add	VERB
cana-2319	51	9	the	the	DET
cana-2319	51	10	same	same	ADJ
cana-2319	51	11	equation	equation	NOUN
cana-2319	51	12	to	to	ADP
cana-2319	51	13	the	the	DET
cana-2319	51	14	system	system	NOUN
cana-2319	51	15	2.1	2.1	NUM
cana-2319	51	16	.	.	PUNCT
cana-2319	52	1	for	for	ADP
cana-2319	52	2	the	the	DET
cana-2319	52	3	second	second	ADJ
cana-2319	52	4	reidemeister	reidemeister	NOUN
cana-2319	52	5	move	move	NOUN
cana-2319	52	6	,	,	PUNCT
cana-2319	52	7	two	two	NUM
cana-2319	52	8	new	new	ADJ
cana-2319	52	9	crossings	crossing	NOUN
cana-2319	52	10	are	be	AUX
cana-2319	52	11	introduced	introduce	VERB
cana-2319	52	12	to	to	ADP
cana-2319	52	13	any	any	DET
cana-2319	52	14	two	two	NUM
cana-2319	52	15	arcs	arc	NOUN
cana-2319	52	16	say	say	VERB
cana-2319	52	17	,	,	PUNCT
cana-2319	52	18	𝑥𝑖	𝑥𝑖	PROPN
cana-2319	52	19	and	and	CCONJ
cana-2319	52	20	𝑥𝑘	𝑥𝑘	X
cana-2319	52	21	as	as	ADP
cana-2319	52	22	in	in	ADP
cana-2319	52	23	figure	figure	NOUN
cana-2319	52	24	4.2	4.2	NUM
cana-2319	52	25	.	.	PUNCT
cana-2319	53	1	again	again	ADV
cana-2319	53	2	,	,	PUNCT
cana-2319	53	3	keeping	keep	VERB
cana-2319	53	4	the	the	DET
cana-2319	53	5	diagram	diagram	NOUN
cana-2319	53	6	fixed	fix	VERB
cana-2319	53	7	except	except	SCONJ
cana-2319	53	8	this	this	DET
cana-2319	53	9	move	move	NOUN
cana-2319	53	10	at	at	ADP
cana-2319	53	11	the	the	DET
cana-2319	53	12	arcs	arc	NOUN
cana-2319	53	13	𝑥𝑗	𝑥𝑗	PROPN
cana-2319	53	14	and	and	CCONJ
cana-2319	53	15	𝑥𝑘.	𝑥𝑘.	ADP
cana-2319	53	16	the	the	DET
cana-2319	53	17	contribution	contribution	NOUN
cana-2319	53	18	of	of	ADP
cana-2319	53	19	this	this	DET
cana-2319	53	20	move	move	NOUN
cana-2319	53	21	to	to	ADP
cana-2319	53	22	the	the	DET
cana-2319	53	23	system	system	NOUN
cana-2319	53	24	2.1	2.1	NUM
cana-2319	53	25	is	be	AUX
cana-2319	53	26	given	give	VERB
cana-2319	53	27	by	by	ADP
cana-2319	53	28	adding	add	VERB
cana-2319	53	29	two	two	NUM
cana-2319	53	30	equations	equation	NOUN
cana-2319	53	31	of	of	ADP
cana-2319	53	32	the	the	DET
cana-2319	53	33	form	form	NOUN
cana-2319	53	34	:	:	PUNCT
cana-2319	53	35	(	(	PUNCT
cana-2319	53	36	1−	1−	X
cana-2319	53	37	𝑡)𝑥𝑖	𝑡)𝑥𝑖	ADJ
cana-2319	53	38	+	+	CCONJ
cana-2319	53	39	𝑡𝑥𝑖	𝑡𝑥𝑖	NOUN
cana-2319	53	40	=	=	PUNCT
cana-2319	53	41	𝑥𝑖.	𝑥𝑖.	ADP
cana-2319	53	42	each	each	DET
cana-2319	53	43	one	one	NUM
cana-2319	53	44	of	of	ADP
cana-2319	53	45	these	these	DET
cana-2319	53	46	two	two	NUM
cana-2319	53	47	equations	equation	NOUN
cana-2319	53	48	can	can	AUX
cana-2319	53	49	be	be	AUX
cana-2319	53	50	rewritten	rewrite	VERB
cana-2319	53	51	in	in	ADP
cana-2319	53	52	the	the	DET
cana-2319	53	53	form	form	NOUN
cana-2319	53	54	:	:	PUNCT
cana-2319	53	55	(	(	PUNCT
cana-2319	53	56	1−	1−	X
cana-2319	53	57	𝑡)𝑥𝑖	𝑡)𝑥𝑖	ADJ
cana-2319	54	1	+	+	NUM
cana-2319	54	2	𝑡𝑥𝑖	𝑡𝑥𝑖	ADJ
cana-2319	54	3	=	=	SYM
cana-2319	54	4	0	0	NUM
cana-2319	54	5	,	,	PUNCT
cana-2319	54	6	which	which	PRON
cana-2319	54	7	do	do	AUX
cana-2319	54	8	not	not	PART
cana-2319	54	9	affect	affect	VERB
cana-2319	54	10	the	the	DET
cana-2319	54	11	solution	solution	NOUN
cana-2319	54	12	of	of	ADP
cana-2319	54	13	the	the	DET
cana-2319	54	14	system	system	NOUN
cana-2319	54	15	2.1	2.1	NUM
cana-2319	54	16	as	as	SCONJ
cana-2319	54	17	we	we	PRON
cana-2319	54	18	mentioned	mention	VERB
cana-2319	54	19	in	in	ADP
cana-2319	54	20	section	section	NOUN
cana-2319	54	21	3	3	NUM
cana-2319	54	22	.	.	PUNCT
cana-2319	55	1	so	so	ADV
cana-2319	55	2	,	,	PUNCT
cana-2319	55	3	∇𝐾(𝑡	∇𝐾(𝑡	X
cana-2319	55	4	)	)	PUNCT
cana-2319	55	5	is	be	AUX
cana-2319	55	6	unchanged	unchanged	ADJ
cana-2319	55	7	under	under	ADP
cana-2319	55	8	the	the	DET
cana-2319	55	9	second	second	ADJ
cana-2319	55	10	reidemeister	reidemeister	NOUN
cana-2319	55	11	communications	communication	NOUN
cana-2319	55	12	on	on	ADP
cana-2319	55	13	applied	apply	VERB
cana-2319	55	14	nonlinear	nonlinear	ADJ
cana-2319	55	15	analysis	analysis	NOUN
cana-2319	55	16	issn	issn	NOUN
cana-2319	55	17	:	:	PUNCT
cana-2319	55	18	1074	1074	NUM
cana-2319	55	19	-	-	PUNCT
cana-2319	55	20	133x	133x	NUM
cana-2319	55	21	vol	vol	NOUN
cana-2319	55	22	32	32	NUM
cana-2319	55	23	no	no	NOUN
cana-2319	55	24	.	.	PUNCT
cana-2319	56	1	1s	1s	NUM
cana-2319	56	2	(	(	PUNCT
cana-2319	56	3	2025	2025	NUM
cana-2319	56	4	)	)	PUNCT
cana-2319	56	5	555	555	NUM
cana-2319	56	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2319	56	7	move	move	NOUN
cana-2319	56	8	.	.	PUNCT
cana-2319	57	1	note	note	NOUN
cana-2319	57	2	.	.	PUNCT
cana-2319	58	1	the	the	DET
cana-2319	58	2	other	other	ADJ
cana-2319	58	3	position	position	NOUN
cana-2319	58	4	of	of	ADP
cana-2319	58	5	the	the	DET
cana-2319	58	6	second	second	ADJ
cana-2319	58	7	reidemeister	reidemeister	NOUN
cana-2319	58	8	move	move	NOUN
cana-2319	58	9	will	will	AUX
cana-2319	58	10	add	add	VERB
cana-2319	58	11	the	the	DET
cana-2319	58	12	same	same	ADJ
cana-2319	58	13	equations	equation	NOUN
cana-2319	58	14	to	to	ADP
cana-2319	58	15	the	the	DET
cana-2319	58	16	system	system	NOUN
cana-2319	58	17	2.1	2.1	NUM
cana-2319	58	18	.	.	PUNCT
cana-2319	59	1	for	for	ADP
cana-2319	59	2	the	the	DET
cana-2319	59	3	third	third	ADJ
cana-2319	59	4	reidemeister	reidemeister	NOUN
cana-2319	59	5	move	move	NOUN
cana-2319	59	6	,	,	PUNCT
cana-2319	59	7	the	the	DET
cana-2319	59	8	contribution	contribution	NOUN
cana-2319	59	9	of	of	ADP
cana-2319	59	10	the	the	DET
cana-2319	59	11	three	three	NUM
cana-2319	59	12	crossings	crossing	NOUN
cana-2319	59	13	before	before	ADP
cana-2319	59	14	and	and	CCONJ
cana-2319	59	15	after	after	SCONJ
cana-2319	59	16	the	the	DET
cana-2319	59	17	move	move	NOUN
cana-2319	59	18	are	be	AUX
cana-2319	59	19	the	the	DET
cana-2319	59	20	same	same	ADJ
cana-2319	59	21	,	,	PUNCT
cana-2319	59	22	so	so	ADV
cana-2319	59	23	∇𝐾(𝑡	∇𝐾(𝑡	X
cana-2319	59	24	)	)	PUNCT
cana-2319	59	25	is	be	AUX
cana-2319	59	26	unchanged	unchanged	ADJ
cana-2319	59	27	under	under	ADP
cana-2319	59	28	the	the	DET
cana-2319	59	29	third	third	ADJ
cana-2319	59	30	reidemeister	reidemeister	NOUN
cana-2319	59	31	move	move	NOUN
cana-2319	59	32	.	.	PUNCT
cana-2319	60	1	finally	finally	ADV
cana-2319	60	2	,	,	PUNCT
cana-2319	60	3	we	we	PRON
cana-2319	60	4	point	point	VERB
cana-2319	60	5	out	out	ADP
cana-2319	60	6	that	that	SCONJ
cana-2319	60	7	we	we	PRON
cana-2319	60	8	used	use	VERB
cana-2319	60	9	the	the	DET
cana-2319	60	10	first	first	ADJ
cana-2319	60	11	and	and	CCONJ
cana-2319	60	12	the	the	DET
cana-2319	60	13	second	second	ADJ
cana-2319	60	14	reidemeister	reidemeister	NOUN
cana-2319	60	15	moves	move	NOUN
cana-2319	60	16	as	as	ADP
cana-2319	60	17	introducing	introduce	VERB
cana-2319	60	18	a	a	DET
cana-2319	60	19	twist	twist	NOUN
cana-2319	60	20	and	and	CCONJ
cana-2319	60	21	two	two	NUM
cana-2319	60	22	new	new	ADJ
cana-2319	60	23	crossings	crossing	NOUN
cana-2319	60	24	,	,	PUNCT
cana-2319	60	25	respectively	respectively	ADV
cana-2319	60	26	,	,	PUNCT
cana-2319	60	27	not	not	PART
cana-2319	60	28	removing	remove	VERB
cana-2319	60	29	them	they	PRON
cana-2319	60	30	.	.	PUNCT
cana-2319	61	1	now	now	ADV
cana-2319	61	2	we	we	PRON
cana-2319	61	3	will	will	AUX
cana-2319	61	4	treat	treat	VERB
cana-2319	61	5	the	the	DET
cana-2319	61	6	case	case	NOUN
cana-2319	61	7	of	of	ADP
cana-2319	61	8	removing	remove	VERB
cana-2319	61	9	.	.	PUNCT
cana-2319	62	1	since	since	SCONJ
cana-2319	62	2	we	we	PRON
cana-2319	62	3	have	have	AUX
cana-2319	62	4	proved	prove	VERB
cana-2319	62	5	that	that	SCONJ
cana-2319	62	6	after	after	ADP
cana-2319	62	7	introducing	introduce	VERB
cana-2319	62	8	a	a	DET
cana-2319	62	9	twist	twist	NOUN
cana-2319	62	10	or	or	CCONJ
cana-2319	62	11	two	two	NUM
cana-2319	62	12	crossings	crossing	NOUN
cana-2319	62	13	,	,	PUNCT
cana-2319	62	14	the	the	DET
cana-2319	62	15	alexander	alexander	PROPN
cana-2319	62	16	polynomial	polynomial	PROPN
cana-2319	62	17	∇𝐾(𝑡	∇𝐾(𝑡	X
cana-2319	62	18	)	)	PUNCT
cana-2319	62	19	does	do	AUX
cana-2319	62	20	not	not	PART
cana-2319	62	21	change	change	VERB
cana-2319	62	22	then	then	ADV
cana-2319	62	23	if	if	SCONJ
cana-2319	62	24	we	we	PRON
cana-2319	62	25	take	take	VERB
cana-2319	62	26	the	the	DET
cana-2319	62	27	last	last	ADJ
cana-2319	62	28	diagram	diagram	NOUN
cana-2319	62	29	after	after	ADP
cana-2319	62	30	introducing	introduce	VERB
cana-2319	62	31	a	a	DET
cana-2319	62	32	twist	twist	NOUN
cana-2319	62	33	or	or	CCONJ
cana-2319	62	34	two	two	NUM
cana-2319	62	35	new	new	ADJ
cana-2319	62	36	crossings	crossing	NOUN
cana-2319	62	37	as	as	ADP
cana-2319	62	38	the	the	DET
cana-2319	62	39	original	original	ADJ
cana-2319	62	40	stage	stage	NOUN
cana-2319	62	41	and	and	CCONJ
cana-2319	62	42	we	we	PRON
cana-2319	62	43	remove	remove	VERB
cana-2319	62	44	a	a	DET
cana-2319	62	45	twist	twist	NOUN
cana-2319	62	46	or	or	CCONJ
cana-2319	62	47	two	two	NUM
cana-2319	62	48	crossings	crossing	NOUN
cana-2319	62	49	by	by	ADP
cana-2319	62	50	the	the	DET
cana-2319	62	51	first	first	ADJ
cana-2319	62	52	or	or	CCONJ
cana-2319	62	53	the	the	DET
cana-2319	62	54	second	second	ADJ
cana-2319	62	55	reidemeister	reidemeister	NOUN
cana-2319	62	56	move	move	NOUN
cana-2319	62	57	,	,	PUNCT
cana-2319	62	58	respectively	respectively	ADV
cana-2319	62	59	,	,	PUNCT
cana-2319	62	60	we	we	PRON
cana-2319	62	61	return	return	VERB
cana-2319	62	62	to	to	ADP
cana-2319	62	63	the	the	DET
cana-2319	62	64	main	main	ADJ
cana-2319	62	65	diagram	diagram	NOUN
cana-2319	62	66	which	which	PRON
cana-2319	62	67	has	have	VERB
cana-2319	62	68	the	the	DET
cana-2319	62	69	same	same	ADJ
cana-2319	62	70	alexander	alexander	NOUN
cana-2319	62	71	polynomial	polynomial	NOUN
cana-2319	62	72	,	,	PUNCT
cana-2319	62	73	∇𝐾(𝑡	∇𝐾(𝑡	NUM
cana-2319	62	74	)	)	PUNCT
cana-2319	62	75	.	.	PUNCT
cana-2319	63	1	5	5	X
cana-2319	63	2	.	.	X
cana-2319	63	3	a	a	DET
cana-2319	63	4	connection	connection	NOUN
cana-2319	63	5	with	with	ADP
cana-2319	63	6	fox	fox	NOUN
cana-2319	63	7	coloring	coloring	NOUN
cana-2319	63	8	in	in	ADP
cana-2319	63	9	[	[	X
cana-2319	63	10	6	6	NUM
cana-2319	63	11	]	]	PUNCT
cana-2319	63	12	,	,	PUNCT
cana-2319	63	13	fox	fox	PROPN
cana-2319	63	14	defined	define	VERB
cana-2319	63	15	tricolorability	tricolorability	NOUN
cana-2319	63	16	or	or	CCONJ
cana-2319	63	17	3	3	NOUN
cana-2319	63	18	-	-	PUNCT
cana-2319	63	19	coloring	coloring	NOUN
cana-2319	63	20	of	of	ADP
cana-2319	63	21	a	a	DET
cana-2319	63	22	knot	knot	ADJ
cana-2319	63	23	diagram	diagram	NOUN
cana-2319	63	24	as	as	ADP
cana-2319	63	25	a	a	DET
cana-2319	63	26	coloring	coloring	NOUN
cana-2319	63	27	of	of	ADP
cana-2319	63	28	one	one	NUM
cana-2319	63	29	of	of	ADP
cana-2319	63	30	its	its	PRON
cana-2319	63	31	diagrams	diagram	NOUN
cana-2319	63	32	with	with	ADP
cana-2319	63	33	three	three	NUM
cana-2319	63	34	colors	color	NOUN
cana-2319	63	35	such	such	ADJ
cana-2319	63	36	that	that	PRON
cana-2319	63	37	:	:	PUNCT
cana-2319	63	38	(	(	PUNCT
cana-2319	63	39	i	i	NOUN
cana-2319	63	40	)	)	PUNCT
cana-2319	63	41	at	at	ADP
cana-2319	63	42	least	least	ADJ
cana-2319	63	43	two	two	NUM
cana-2319	63	44	different	different	ADJ
cana-2319	63	45	colors	color	NOUN
cana-2319	63	46	are	be	AUX
cana-2319	63	47	used	use	VERB
cana-2319	63	48	.	.	PUNCT
cana-2319	64	1	(	(	PUNCT
cana-2319	64	2	ii	ii	NOUN
cana-2319	64	3	)	)	PUNCT
cana-2319	64	4	at	at	ADP
cana-2319	64	5	each	each	DET
cana-2319	64	6	crossing	crossing	NOUN
cana-2319	64	7	,	,	PUNCT
cana-2319	64	8	the	the	DET
cana-2319	64	9	three	three	NUM
cana-2319	64	10	arcs	arc	NOUN
cana-2319	64	11	are	be	AUX
cana-2319	64	12	of	of	ADP
cana-2319	64	13	the	the	DET
cana-2319	64	14	same	same	ADJ
cana-2319	64	15	color	color	NOUN
cana-2319	64	16	or	or	CCONJ
cana-2319	64	17	all	all	PRON
cana-2319	64	18	are	be	AUX
cana-2319	64	19	of	of	ADP
cana-2319	64	20	distinct	distinct	ADJ
cana-2319	64	21	colors	color	NOUN
cana-2319	64	22	.	.	PUNCT
cana-2319	65	1	note	note	VERB
cana-2319	65	2	.	.	PUNCT
cana-2319	66	1	this	this	DET
cana-2319	66	2	definition	definition	NOUN
cana-2319	66	3	does	do	AUX
cana-2319	66	4	not	not	PART
cana-2319	66	5	require	require	VERB
cana-2319	66	6	the	the	DET
cana-2319	66	7	given	give	VERB
cana-2319	66	8	diagram	diagram	NOUN
cana-2319	66	9	to	to	PART
cana-2319	66	10	be	be	AUX
cana-2319	66	11	oriented	orient	VERB
cana-2319	66	12	.	.	PUNCT
cana-2319	67	1	furthermore	furthermore	ADV
cana-2319	67	2	,	,	PUNCT
cana-2319	67	3	it	it	PRON
cana-2319	67	4	is	be	AUX
cana-2319	67	5	proved	prove	VERB
cana-2319	67	6	that	that	SCONJ
cana-2319	67	7	3	3	NUM
cana-2319	67	8	-	-	PUNCT
cana-2319	67	9	coloring	coloring	NOUN
cana-2319	67	10	is	be	AUX
cana-2319	67	11	a	a	DET
cana-2319	67	12	knot	knot	NOUN
cana-2319	67	13	invariant	invariant	ADJ
cana-2319	67	14	.	.	PUNCT
cana-2319	68	1	it	it	PRON
cana-2319	68	2	seems	seem	VERB
cana-2319	68	3	that	that	SCONJ
cana-2319	68	4	3	3	NUM
cana-2319	68	5	-	-	PUNCT
cana-2319	68	6	coloring	coloring	NOUN
cana-2319	68	7	is	be	AUX
cana-2319	68	8	not	not	PART
cana-2319	68	9	numerical	numerical	ADJ
cana-2319	68	10	knot	knot	NOUN
cana-2319	68	11	invariant	invariant	ADJ
cana-2319	68	12	,	,	PUNCT
cana-2319	68	13	but	but	CCONJ
cana-2319	68	14	this	this	DET
cana-2319	68	15	view	view	NOUN
cana-2319	68	16	may	may	AUX
cana-2319	68	17	change	change	VERB
cana-2319	68	18	when	when	SCONJ
cana-2319	68	19	we	we	PRON
cana-2319	68	20	connect	connect	VERB
cana-2319	68	21	3	3	NUM
cana-2319	68	22	-	-	PUNCT
cana-2319	68	23	coloring	coloring	NOUN
cana-2319	68	24	with	with	ADP
cana-2319	68	25	the	the	DET
cana-2319	68	26	system	system	NOUN
cana-2319	68	27	of	of	ADP
cana-2319	68	28	homogeneous	homogeneous	ADJ
cana-2319	68	29	linear	linear	PROPN
cana-2319	68	30	equations	equation	NOUN
cana-2319	68	31	2.1	2.1	NUM
cana-2319	68	32	,	,	PUNCT
cana-2319	68	33	where	where	SCONJ
cana-2319	68	34	3	3	NUM
cana-2319	68	35	-	-	PUNCT
cana-2319	68	36	coloring	coloring	NOUN
cana-2319	68	37	becomes	become	VERB
cana-2319	68	38	a	a	DET
cana-2319	68	39	linear	linear	ADJ
cana-2319	68	40	algebra	algebra	NOUN
cana-2319	68	41	problem	problem	NOUN
cana-2319	68	42	ignoring	ignore	VERB
cana-2319	68	43	the	the	DET
cana-2319	68	44	orientation	orientation	NOUN
cana-2319	68	45	which	which	PRON
cana-2319	68	46	is	be	AUX
cana-2319	68	47	necessary	necessary	ADJ
cana-2319	68	48	for	for	ADP
cana-2319	68	49	the	the	DET
cana-2319	68	50	system	system	NOUN
cana-2319	68	51	2.1	2.1	NUM
cana-2319	68	52	.	.	PUNCT
cana-2319	69	1	recall	recall	VERB
cana-2319	69	2	one	one	NUM
cana-2319	69	3	equation	equation	NOUN
cana-2319	69	4	from	from	ADP
cana-2319	69	5	the	the	DET
cana-2319	69	6	system	system	NOUN
cana-2319	69	7	2.1	2.1	NUM
cana-2319	69	8	say	say	NOUN
cana-2319	69	9	,	,	PUNCT
cana-2319	69	10	𝑡𝑥𝑖	𝑡𝑥𝑖	NOUN
cana-2319	69	11	+	+	CCONJ
cana-2319	69	12	(	(	PUNCT
cana-2319	69	13	1−	1−	NUM
cana-2319	69	14	𝑡)𝑥𝑖	𝑡)𝑥𝑖	PROPN
cana-2319	69	15	=	=	PUNCT
cana-2319	70	1	𝑥𝑘.	𝑥𝑘.	ADJ
cana-2319	70	2	if	if	SCONJ
cana-2319	70	3	we	we	PRON
cana-2319	70	4	put	put	VERB
cana-2319	70	5	𝑡	𝑡	NOUN
cana-2319	70	6	=	=	NOUN
cana-2319	70	7	−1	−1	NOUN
cana-2319	70	8	we	we	PRON
cana-2319	70	9	get	get	VERB
cana-2319	70	10	2𝑥𝑖	2𝑥𝑖	ADJ
cana-2319	70	11	−	−	PROPN
cana-2319	70	12	𝑥𝑖	𝑥𝑖	NUM
cana-2319	70	13	−	−	PROPN
cana-2319	70	14	𝑥𝑘	𝑥𝑘	NOUN
cana-2319	70	15	=	=	NOUN
cana-2319	70	16	0	0	PROPN
cana-2319	70	17	.	.	PUNCT
cana-2319	71	1	this	this	DET
cana-2319	71	2	equation	equation	NOUN
cana-2319	71	3	is	be	AUX
cana-2319	71	4	called	call	VERB
cana-2319	71	5	fox	fox	PROPN
cana-2319	71	6	coloring	coloring	NOUN
cana-2319	71	7	condition	condition	NOUN
cana-2319	71	8	of	of	ADP
cana-2319	71	9	the	the	DET
cana-2319	71	10	crossing	crossing	NOUN
cana-2319	71	11	with	with	ADP
cana-2319	71	12	an	an	DET
cana-2319	71	13	upper	upper	ADJ
cana-2319	71	14	arc	arc	NOUN
cana-2319	71	15	𝑥𝑗	𝑥𝑗	PROPN
cana-2319	71	16	and	and	CCONJ
cana-2319	71	17	two	two	NUM
cana-2319	71	18	lower	low	ADJ
cana-2319	71	19	arcs	arc	NOUN
cana-2319	71	20	𝑥𝑖	𝑥𝑖	X
cana-2319	71	21	and	and	CCONJ
cana-2319	71	22	𝑥𝑘.	𝑥𝑘.	ADV
cana-2319	71	23	so	so	ADV
cana-2319	71	24	,	,	PUNCT
cana-2319	71	25	we	we	PRON
cana-2319	71	26	can	can	AUX
cana-2319	71	27	assign	assign	VERB
cana-2319	71	28	any	any	DET
cana-2319	71	29	given	give	VERB
cana-2319	71	30	knot	knot	ADJ
cana-2319	71	31	diagram	diagram	NOUN
cana-2319	71	32	with	with	ADP
cana-2319	71	33	a	a	DET
cana-2319	71	34	new	new	ADJ
cana-2319	71	35	system	system	NOUN
cana-2319	71	36	of	of	ADP
cana-2319	71	37	linear	linear	ADJ
cana-2319	71	38	homogeneous	homogeneous	ADJ
cana-2319	71	39	equations	equation	NOUN
cana-2319	71	40	over	over	ADP
cana-2319	71	41	ℤ	ℤ	PROPN
cana-2319	71	42	called	call	VERB
cana-2319	71	43	the	the	DET
cana-2319	71	44	coloring	coloring	NOUN
cana-2319	71	45	system	system	NOUN
cana-2319	71	46	corresponds	correspond	VERB
cana-2319	71	47	to	to	ADP
cana-2319	71	48	the	the	DET
cana-2319	71	49	system	system	NOUN
cana-2319	71	50	2.1	2.1	NUM
cana-2319	71	51	with	with	ADP
cana-2319	71	52	𝑡	𝑡	PROPN
cana-2319	71	53	=	=	SYM
cana-2319	71	54	−1	−1	NOUN
cana-2319	71	55	,	,	PUNCT
cana-2319	71	56	which	which	PRON
cana-2319	71	57	is	be	AUX
cana-2319	71	58	always	always	ADV
cana-2319	71	59	consistent	consistent	ADJ
cana-2319	71	60	.	.	PUNCT
cana-2319	72	1	if	if	SCONJ
cana-2319	72	2	we	we	PRON
cana-2319	72	3	consider	consider	VERB
cana-2319	72	4	this	this	DET
cana-2319	72	5	new	new	ADJ
cana-2319	72	6	system	system	NOUN
cana-2319	72	7	of	of	ADP
cana-2319	72	8	homogeneous	homogeneous	ADJ
cana-2319	72	9	linear	linear	PROPN
cana-2319	72	10	equations	equation	NOUN
cana-2319	72	11	over	over	ADP
cana-2319	72	12	ℤ3	ℤ3	PROPN
cana-2319	72	13	,	,	PUNCT
cana-2319	72	14	the	the	DET
cana-2319	72	15	3	3	NUM
cana-2319	72	16	-	-	PUNCT
cana-2319	72	17	coloring	color	VERB
cana-2319	72	18	problem	problem	NOUN
cana-2319	72	19	becomes	become	VERB
cana-2319	72	20	the	the	DET
cana-2319	72	21	problem	problem	NOUN
cana-2319	72	22	of	of	ADP
cana-2319	72	23	the	the	DET
cana-2319	72	24	existence	existence	NOUN
cana-2319	72	25	or	or	CCONJ
cana-2319	72	26	non	non	ADJ
cana-2319	72	27	-	-	NOUN
cana-2319	72	28	existence	existence	NOUN
cana-2319	72	29	of	of	ADP
cana-2319	72	30	nontrivial	nontrivial	ADJ
cana-2319	72	31	solutions	solution	NOUN
cana-2319	72	32	of	of	ADP
cana-2319	72	33	the	the	DET
cana-2319	72	34	system	system	NOUN
cana-2319	72	35	𝐴𝑋	𝐴𝑋	PROPN
cana-2319	72	36	=	=	SYM
cana-2319	72	37	0⃑	0⃑	NOUN
cana-2319	72	38	,	,	PUNCT
cana-2319	72	39	where	where	SCONJ
cana-2319	72	40	𝐴	𝐴	PROPN
cana-2319	72	41	is	be	AUX
cana-2319	72	42	any	any	DET
cana-2319	72	43	3	3	NUM
cana-2319	72	44	×	×	NOUN
cana-2319	72	45	3	3	NUM
cana-2319	72	46	-	-	PUNCT
cana-2319	72	47	matrix	matrix	NOUN
cana-2319	72	48	obtained	obtain	VERB
cana-2319	72	49	by	by	ADP
cana-2319	72	50	omitting	omit	VERB
cana-2319	72	51	some	some	DET
cana-2319	72	52	rows	row	NOUN
cana-2319	72	53	and	and	CCONJ
cana-2319	72	54	the	the	DET
cana-2319	72	55	same	same	ADJ
cana-2319	72	56	number	number	NOUN
cana-2319	72	57	of	of	ADP
cana-2319	72	58	columns	column	NOUN
cana-2319	72	59	from	from	ADP
cana-2319	72	60	the	the	DET
cana-2319	72	61	coefficients	coefficient	NOUN
cana-2319	72	62	matrix	matrix	NOUN
cana-2319	72	63	of	of	ADP
cana-2319	72	64	the	the	DET
cana-2319	72	65	coloring	coloring	NOUN
cana-2319	72	66	system	system	NOUN
cana-2319	72	67	.	.	PUNCT
cana-2319	73	1	now	now	ADV
cana-2319	73	2	why	why	SCONJ
cana-2319	73	3	not	not	PART
cana-2319	73	4	?	?	PUNCT
cana-2319	74	1	we	we	PRON
cana-2319	74	2	can	can	AUX
cana-2319	74	3	collect	collect	VERB
cana-2319	74	4	what	what	PRON
cana-2319	74	5	we	we	PRON
cana-2319	74	6	did	do	VERB
cana-2319	74	7	in	in	ADP
cana-2319	74	8	the	the	DET
cana-2319	74	9	proof	proof	NOUN
cana-2319	74	10	of	of	ADP
cana-2319	74	11	the	the	DET
cana-2319	74	12	main	main	ADJ
cana-2319	74	13	theorem	theorem	NOUN
cana-2319	74	14	in	in	ADP
cana-2319	74	15	section	section	NOUN
cana-2319	74	16	4	4	NUM
cana-2319	74	17	with	with	ADP
cana-2319	74	18	what	what	PRON
cana-2319	74	19	we	we	PRON
cana-2319	74	20	mentioned	mention	VERB
cana-2319	74	21	about	about	ADP
cana-2319	74	22	the	the	DET
cana-2319	74	23	3	3	NUM
cana-2319	74	24	-	-	PUNCT
cana-2319	74	25	coloring	color	VERB
cana-2319	74	26	property	property	NOUN
cana-2319	74	27	that	that	PRON
cana-2319	74	28	it	it	PRON
cana-2319	74	29	depends	depend	VERB
cana-2319	74	30	on	on	ADP
cana-2319	74	31	the	the	DET
cana-2319	74	32	solutions	solution	NOUN
cana-2319	74	33	of	of	ADP
cana-2319	74	34	a	a	DET
cana-2319	74	35	system	system	NOUN
cana-2319	74	36	obtained	obtain	VERB
cana-2319	74	37	from	from	ADP
cana-2319	74	38	the	the	DET
cana-2319	74	39	system	system	NOUN
cana-2319	74	40	2.1	2.1	NUM
cana-2319	74	41	of	of	ADP
cana-2319	74	42	the	the	DET
cana-2319	74	43	given	give	VERB
cana-2319	74	44	knot	knot	NOUN
cana-2319	74	45	diagram	diagram	NOUN
cana-2319	74	46	to	to	ADP
cana-2319	74	47	state	state	VERB
cana-2319	74	48	that	that	SCONJ
cana-2319	74	49	reidemeister	reidemeister	ADJ
cana-2319	74	50	moves	move	NOUN
cana-2319	74	51	do	do	AUX
cana-2319	74	52	not	not	PART
cana-2319	74	53	change	change	VERB
cana-2319	74	54	the	the	DET
cana-2319	74	55	solutions	solution	NOUN
cana-2319	74	56	space	space	NOUN
cana-2319	74	57	of	of	ADP
cana-2319	74	58	the	the	DET
cana-2319	74	59	system	system	NOUN
cana-2319	74	60	2.1	2.1	NUM
cana-2319	74	61	and	and	CCONJ
cana-2319	74	62	then	then	ADV
cana-2319	74	63	3	3	X
cana-2319	74	64	-	-	PUNCT
cana-2319	74	65	coloring	coloring	NOUN
cana-2319	74	66	of	of	ADP
cana-2319	74	67	the	the	DET
cana-2319	74	68	given	give	VERB
cana-2319	74	69	diagram	diagram	NOUN
cana-2319	74	70	.	.	PUNCT
cana-2319	75	1	this	this	PRON
cana-2319	75	2	can	can	AUX
cana-2319	75	3	be	be	AUX
cana-2319	75	4	considered	consider	VERB
cana-2319	75	5	as	as	ADP
cana-2319	75	6	a	a	DET
cana-2319	75	7	proof	proof	NOUN
cana-2319	75	8	of	of	ADP
cana-2319	75	9	a	a	DET
cana-2319	75	10	similar	similar	ADJ
cana-2319	75	11	theorem	theorem	NOUN
cana-2319	75	12	to	to	ADP
cana-2319	75	13	the	the	DET
cana-2319	75	14	theorem	theorem	NOUN
cana-2319	75	15	in	in	ADP
cana-2319	75	16	section	section	NOUN
cana-2319	75	17	4	4	NUM
cana-2319	75	18	that	that	PRON
cana-2319	75	19	is	be	AUX
cana-2319	75	20	3	3	NUM
cana-2319	75	21	-	-	PUNCT
cana-2319	75	22	coloring	coloring	NOUN
cana-2319	75	23	is	be	AUX
cana-2319	75	24	a	a	DET
cana-2319	75	25	knot	knot	NOUN
cana-2319	75	26	invariant	invariant	ADJ
cana-2319	75	27	.	.	PUNCT
cana-2319	75	28	applications	application	NOUN
cana-2319	75	29	.	.	PUNCT
cana-2319	76	1	5.1	5.1	NUM
cana-2319	76	2	the	the	DET
cana-2319	76	3	trefoil	trefoil	NOUN
cana-2319	76	4	:	:	PUNCT
cana-2319	76	5	we	we	PRON
cana-2319	76	6	can	can	AUX
cana-2319	76	7	return	return	VERB
cana-2319	76	8	to	to	ADP
cana-2319	76	9	the	the	DET
cana-2319	76	10	obtained	obtain	VERB
cana-2319	76	11	system	system	NOUN
cana-2319	76	12	for	for	ADP
cana-2319	76	13	the	the	DET
cana-2319	76	14	trefoil	trefoil	NOUN
cana-2319	76	15	in	in	ADP
cana-2319	76	16	section	section	NOUN
cana-2319	76	17	2	2	NUM
cana-2319	76	18	to	to	PART
cana-2319	76	19	solve	solve	VERB
cana-2319	76	20	the	the	DET
cana-2319	76	21	3	3	NUM
cana-2319	76	22	-	-	PUNCT
cana-2319	76	23	coloring	color	VERB
cana-2319	76	24	system	system	NOUN
cana-2319	76	25	that	that	PRON
cana-2319	76	26	is	be	AUX
cana-2319	76	27	[	[	PUNCT
cana-2319	76	28	2	2	NUM
cana-2319	76	29	−1	−1	NOUN
cana-2319	76	30	−1	−1	NOUN
cana-2319	76	31	−1	−1	NOUN
cana-2319	76	32	2	2	NUM
cana-2319	76	33	−1	−1	NOUN
cana-2319	76	34	−1	−1	NOUN
cana-2319	76	35	−1	−1	NOUN
cana-2319	76	36	2	2	NUM
cana-2319	76	37	]	]	PUNCT
cana-2319	76	38	[	[	PUNCT
cana-2319	76	39	𝑥	𝑥	X
cana-2319	76	40	𝑦	𝑦	NOUN
cana-2319	76	41	𝑧	𝑧	X
cana-2319	76	42	]	]	PUNCT
cana-2319	76	43	=	=	PUNCT
cana-2319	76	44	[	[	PUNCT
cana-2319	76	45	0	0	NUM
cana-2319	76	46	0	0	NUM
cana-2319	76	47	0	0	NUM
cana-2319	76	48	]	]	PUNCT
cana-2319	76	49	since	since	SCONJ
cana-2319	76	50	the	the	DET
cana-2319	76	51	system	system	NOUN
cana-2319	76	52	has	have	VERB
cana-2319	76	53	non	non	ADJ
cana-2319	76	54	-	-	ADJ
cana-2319	76	55	trivial	trivial	ADJ
cana-2319	76	56	solutions	solution	NOUN
cana-2319	76	57	then	then	ADV
cana-2319	76	58	the	the	DET
cana-2319	76	59	trefoil	trefoil	NOUN
cana-2319	76	60	is	be	AUX
cana-2319	76	61	3	3	NUM
cana-2319	76	62	-	-	PUNCT
cana-2319	76	63	coloring	coloring	NOUN
cana-2319	76	64	.	.	PUNCT
cana-2319	77	1	5.2	5.2	NUM
cana-2319	77	2	the	the	DET
cana-2319	77	3	figure	figure	NOUN
cana-2319	77	4	eight	eight	NUM
cana-2319	77	5	:	:	PUNCT
cana-2319	77	6	given	give	VERB
cana-2319	77	7	any	any	DET
cana-2319	77	8	oriented	orient	VERB
cana-2319	77	9	diagram	diagram	NOUN
cana-2319	77	10	of	of	ADP
cana-2319	77	11	the	the	DET
cana-2319	77	12	figure	figure	NOUN
cana-2319	77	13	eight	eight	NUM
cana-2319	77	14	,	,	PUNCT
cana-2319	77	15	we	we	PRON
cana-2319	77	16	can	can	AUX
cana-2319	77	17	easily	easily	ADV
cana-2319	77	18	check	check	VERB
cana-2319	77	19	that	that	SCONJ
cana-2319	77	20	its	its	PRON
cana-2319	77	21	coloring	coloring	NOUN
cana-2319	77	22	system	system	NOUN
cana-2319	77	23	is	be	AUX
cana-2319	77	24	communications	communication	NOUN
cana-2319	77	25	on	on	ADP
cana-2319	77	26	applied	apply	VERB
cana-2319	77	27	nonlinear	nonlinear	ADJ
cana-2319	77	28	analysis	analysis	NOUN
cana-2319	77	29	issn	issn	NOUN
cana-2319	77	30	:	:	PUNCT
cana-2319	77	31	1074	1074	NUM
cana-2319	77	32	-	-	PUNCT
cana-2319	77	33	133x	133x	NUM
cana-2319	77	34	vol	vol	NOUN
cana-2319	77	35	32	32	NUM
cana-2319	77	36	no	no	NOUN
cana-2319	77	37	.	.	PUNCT
cana-2319	78	1	1s	1s	NUM
cana-2319	78	2	(	(	PUNCT
cana-2319	78	3	2025	2025	NUM
cana-2319	78	4	)	)	PUNCT
cana-2319	78	5	556	556	NUM
cana-2319	78	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2319	78	7	[	[	PUNCT
cana-2319	78	8	2	2	NUM
cana-2319	78	9	−1	−1	NOUN
cana-2319	78	10	0	0	NUM
cana-2319	78	11	−1	−1	NOUN
cana-2319	78	12	0	0	NUM
cana-2319	78	13	2	2	NUM
cana-2319	78	14	−1	−1	NOUN
cana-2319	78	15	−1	−1	NOUN
cana-2319	78	16	−1	−1	NOUN
cana-2319	78	17	−1	−1	NOUN
cana-2319	78	18	−1	−1	NOUN
cana-2319	78	19	0	0	NUM
cana-2319	78	20	2	2	NUM
cana-2319	78	21	−1	−1	NOUN
cana-2319	78	22	0	0	NUM
cana-2319	78	23	2	2	NUM
cana-2319	78	24	]	]	PUNCT
cana-2319	78	25	[	[	PUNCT
cana-2319	78	26	𝑥	𝑥	X
cana-2319	78	27	𝑦	𝑦	NOUN
cana-2319	78	28	𝑧	𝑧	VERB
cana-2319	78	29	𝑤	𝑤	ADP
cana-2319	78	30	]	]	PUNCT
cana-2319	78	31	=	=	PUNCT
cana-2319	79	1	[	[	PUNCT
cana-2319	79	2	0	0	NUM
cana-2319	79	3	0	0	NUM
cana-2319	79	4	0	0	NUM
cana-2319	79	5	0	0	NUM
cana-2319	79	6	]	]	PUNCT
cana-2319	79	7	since	since	SCONJ
cana-2319	79	8	any	any	DET
cana-2319	79	9	3	3	NUM
cana-2319	79	10	×	×	NOUN
cana-2319	79	11	3	3	NUM
cana-2319	79	12	-	-	PUNCT
cana-2319	79	13	matrix	matrix	NOUN
cana-2319	79	14	𝐴	𝐴	PROPN
cana-2319	79	15	obtained	obtain	VERB
cana-2319	79	16	from	from	ADP
cana-2319	79	17	the	the	DET
cana-2319	79	18	coefficient	coefficient	NOUN
cana-2319	79	19	matrix	matrix	NOUN
cana-2319	79	20	of	of	ADP
cana-2319	79	21	the	the	DET
cana-2319	79	22	last	last	ADJ
cana-2319	79	23	system	system	NOUN
cana-2319	79	24	by	by	ADP
cana-2319	79	25	omitting	omit	VERB
cana-2319	79	26	one	one	NUM
cana-2319	79	27	row	row	NOUN
cana-2319	79	28	and	and	CCONJ
cana-2319	79	29	one	one	NUM
cana-2319	79	30	column	column	NOUN
cana-2319	79	31	is	be	AUX
cana-2319	79	32	non	non	ADJ
cana-2319	79	33	-	-	ADJ
cana-2319	79	34	singular	singular	ADJ
cana-2319	79	35	then	then	ADV
cana-2319	79	36	the	the	DET
cana-2319	79	37	system	system	NOUN
cana-2319	79	38	𝐴𝑋	𝐴𝑋	PROPN
cana-2319	79	39	=	=	SYM
cana-2319	79	40	0⃑	0⃑	PROPN
cana-2319	79	41	has	have	VERB
cana-2319	79	42	only	only	ADV
cana-2319	79	43	the	the	DET
cana-2319	79	44	trivial	trivial	ADJ
cana-2319	79	45	solution	solution	NOUN
cana-2319	79	46	for	for	ADP
cana-2319	79	47	each	each	DET
cana-2319	79	48	𝐴.	𝐴.	NOUN
cana-2319	79	49	this	this	PRON
cana-2319	79	50	confirms	confirm	VERB
cana-2319	79	51	that	that	SCONJ
cana-2319	79	52	the	the	DET
cana-2319	79	53	figure	figure	NOUN
cana-2319	79	54	eight	eight	NUM
cana-2319	79	55	is	be	AUX
cana-2319	79	56	not	not	PART
cana-2319	79	57	3	3	NUM
cana-2319	79	58	-	-	PUNCT
cana-2319	79	59	coloring	coloring	NOUN
cana-2319	79	60	.	.	PUNCT
cana-2319	80	1	references	reference	NOUN
cana-2319	80	2	[	[	X
cana-2319	80	3	1	1	NUM
cana-2319	80	4	]	]	PUNCT
cana-2319	80	5	c.	c.	PROPN
cana-2319	80	6	livingston	livingston	PROPN
cana-2319	80	7	,	,	PUNCT
cana-2319	80	8	knot	knot	NOUN
cana-2319	80	9	theory	theory	NOUN
cana-2319	80	10	,	,	PUNCT
cana-2319	80	11	the	the	DET
cana-2319	80	12	mathematical	mathematical	ADJ
cana-2319	80	13	association	association	NOUN
cana-2319	80	14	of	of	ADP
cana-2319	80	15	america	america	PROPN
cana-2319	80	16	,	,	PUNCT
cana-2319	80	17	washington	washington	PROPN
cana-2319	80	18	d.	d.	PROPN
cana-2319	80	19	c	c	PROPN
cana-2319	80	20	,	,	PUNCT
cana-2319	80	21	1993	1993	NUM
cana-2319	80	22	.	.	PUNCT
cana-2319	81	1	[	[	X
cana-2319	81	2	2	2	X
cana-2319	81	3	]	]	PUNCT
cana-2319	81	4	e.	e.	PROPN
cana-2319	81	5	long	long	PROPN
cana-2319	81	6	,	,	PUNCT
cana-2319	81	7	three	three	NUM
cana-2319	81	8	routes	route	NOUN
cana-2319	81	9	to	to	ADP
cana-2319	81	10	alexander	alexander	PROPN
cana-2319	81	11	polynomial	polynomial	PROPN
cana-2319	81	12	,	,	PUNCT
cana-2319	81	13	manchester	manchester	PROPN
cana-2319	81	14	university	university	PROPN
cana-2319	81	15	,	,	PUNCT
cana-2319	81	16	2005	2005	NUM
cana-2319	81	17	.	.	PUNCT
cana-2319	82	1	[	[	X
cana-2319	82	2	3	3	X
cana-2319	82	3	]	]	X
cana-2319	82	4	j.	j.	PROPN
cana-2319	82	5	may	may	PROPN
cana-2319	82	6	,	,	PUNCT
cana-2319	82	7	matrix	matrix	VERB
cana-2319	82	8	representation	representation	NOUN
cana-2319	82	9	of	of	ADP
cana-2319	82	10	knot	knot	NOUN
cana-2319	82	11	and	and	CCONJ
cana-2319	82	12	link	link	NOUN
cana-2319	82	13	groups	group	NOUN
cana-2319	82	14	,	,	PUNCT
cana-2319	82	15	harvey	harvey	PROPN
cana-2319	82	16	mudd	mudd	PROPN
cana-2319	82	17	college	college	PROPN
cana-2319	82	18	,	,	PUNCT
cana-2319	82	19	2006	2006	NUM
cana-2319	82	20	.	.	PUNCT
cana-2319	83	1	[	[	X
cana-2319	83	2	4	4	X
cana-2319	83	3	]	]	PUNCT
cana-2319	83	4	j.	j.	PROPN
cana-2319	83	5	w.	w.	PROPN
cana-2319	83	6	alexander	alexander	PROPN
cana-2319	83	7	,	,	PUNCT
cana-2319	83	8	topological	topological	ADJ
cana-2319	83	9	invariants	invariant	NOUN
cana-2319	83	10	of	of	ADP
cana-2319	83	11	knots	knot	NOUN
cana-2319	83	12	and	and	CCONJ
cana-2319	83	13	links	link	NOUN
cana-2319	83	14	,	,	PUNCT
cana-2319	83	15	transactions	transaction	NOUN
cana-2319	83	16	of	of	ADP
cana-2319	83	17	american	american	PROPN
cana-2319	83	18	mathematical	mathematical	PROPN
cana-2319	83	19	society	society	NOUN
cana-2319	83	20	,	,	PUNCT
cana-2319	83	21	30(2	30(2	NUM
cana-2319	83	22	)	)	PUNCT
cana-2319	83	23	,	,	PUNCT
cana-2319	83	24	1928	1928	NUM
cana-2319	83	25	,	,	PUNCT
cana-2319	83	26	275	275	NUM
cana-2319	83	27	-	-	SYM
cana-2319	83	28	306	306	NUM
cana-2319	83	29	.	.	PUNCT
cana-2319	84	1	[	[	X
cana-2319	84	2	5	5	X
cana-2319	84	3	]	]	PUNCT
cana-2319	84	4	l.	l.	PROPN
cana-2319	84	5	h.	h.	PROPN
cana-2319	84	6	kauffman	kauffman	PROPN
cana-2319	84	7	,	,	PUNCT
cana-2319	84	8	p.	p.	PROPN
cana-2319	84	9	lopes	lopes	PROPN
cana-2319	84	10	,	,	PUNCT
cana-2319	84	11	coloring	color	VERB
cana-2319	84	12	beyond	beyond	ADP
cana-2319	84	13	fox	fox	NOUN
cana-2319	84	14	:	:	PUNCT
cana-2319	84	15	the	the	DET
cana-2319	84	16	other	other	ADJ
cana-2319	84	17	linear	linear	PROPN
cana-2319	84	18	alexander	alexander	PROPN
cana-2319	84	19	quandles	quandles	PROPN
cana-2319	84	20	,	,	PUNCT
cana-2319	84	21	arxiv	arxiv	PROPN
cana-2319	84	22	:	:	PUNCT
cana-2319	84	23	1708	1708	NUM
cana-2319	84	24	.	.	PUNCT
cana-2319	85	1	01932v3[math	01932v3[math	AUX
cana-2319	85	2	.	.	PUNCT
cana-2319	86	1	gt	gt	PROPN
cana-2319	86	2	]	]	PUNCT
cana-2319	86	3	,	,	PUNCT
cana-2319	86	4	april	april	PROPN
cana-2319	86	5	10	10	NUM
cana-2319	86	6	,	,	PUNCT
cana-2319	86	7	2018	2018	NUM
cana-2319	86	8	.	.	PUNCT
cana-2319	87	1	[	[	X
cana-2319	87	2	6	6	NUM
cana-2319	87	3	]	]	PUNCT
cana-2319	87	4	r.	r.	PROPN
cana-2319	87	5	h.	h.	PROPN
cana-2319	87	6	crowell	crowell	PROPN
cana-2319	87	7	and	and	CCONJ
cana-2319	87	8	r.	r.	PROPN
cana-2319	87	9	h.	h.	PROPN
cana-2319	87	10	fox	fox	PROPN
cana-2319	87	11	,	,	PUNCT
cana-2319	87	12	introduction	introduction	NOUN
cana-2319	87	13	to	to	PART
cana-2319	87	14	knot	knot	VERB
cana-2319	87	15	theory	theory	NOUN
cana-2319	87	16	,	,	PUNCT
cana-2319	87	17	ginn	ginn	PROPN
cana-2319	87	18	and	and	CCONJ
cana-2319	87	19	company	company	NOUN
cana-2319	87	20	,	,	PUNCT
cana-2319	87	21	1963	1963	NUM
cana-2319	87	22	.	.	PUNCT
cana-2319	88	1	[	[	X
cana-2319	88	2	7	7	X
cana-2319	88	3	]	]	X
cana-2319	88	4	r.	r.	PROPN
cana-2319	88	5	fox	fox	PROPN
cana-2319	88	6	,	,	PUNCT
cana-2319	88	7	a	a	DET
cana-2319	88	8	quick	quick	ADJ
cana-2319	88	9	trip	trip	NOUN
cana-2319	88	10	through	through	ADP
cana-2319	88	11	knot	knot	NOUN
cana-2319	88	12	theory	theory	NOUN
cana-2319	88	13	,	,	PUNCT
cana-2319	88	14	topology	topology	NOUN
cana-2319	88	15	of	of	ADP
cana-2319	88	16	3	3	NUM
cana-2319	88	17	-	-	PUNCT
cana-2319	88	18	manifolds	manifolds	ADJ
cana-2319	88	19	and	and	CCONJ
cana-2319	88	20	related	related	ADJ
cana-2319	88	21	topics	topic	NOUN
cana-2319	88	22	,	,	PUNCT
cana-2319	88	23	86	86	NUM
cana-2319	88	24	,	,	PUNCT
cana-2319	88	25	1949	1949	NUM
cana-2319	88	26	,	,	PUNCT
cana-2319	88	27	120	120	NUM
cana-2319	88	28	-	-	SYM
cana-2319	88	29	1	1	NUM