@ Appl. Gen. Topol. 21, no. 2 (2020), 171-176 doi:10.4995/agt.2020.10129 c© AGT, UPV, 2020 Structure of symmetry group of some composite links and some applications Yang Liu∗ Shenzhen Technology University, Shenzhen, China (liuyang2@sztu.edu.cn) Communicated by F. Lin Abstract In this paper, we study the symmetry group of a type of composite topological links, such as 221m!2 2 1. We have done a complete analysis on the elements of the symmetric group of this link and show the struc- ture of the group. The results can be generalized to the study of the symmetry group of any composite topological link, and therefore it can be used for the classification of composite topological links, which can also be potentially used to identify synthetics molecules. 2010 MSC: 57Q45; 57M25; 20B30; 20B35; 51H05. Keywords: knot; link; geometric topology; symmetry group; classification of links. 1. Introduction In topology, geometry, and physics, various knots, which are mathematically various embeddings of a circle in the 3-dimensional Euclidean space, have been interesting objects, which have been studied in recent decades (see for instance, [12], [3], and [10]). In particular, knots have been used to construct examples for the study of low-dimensional topology (see for instance, [9]). Two or more knots can make up of a link, which have appeared to be somewhat more interesting as a single knot, because of the combinatorial structure involved. The theory of knots and links has applications in many areas such as physics, biochemistry, ∗This work is partially supported by Shenzhen Municipal Finance for Research. Received 13 September 2019 – Accepted 22 November 2019 http://dx.doi.org/10.4995/agt.2020.10129 Y. Liu and biology, in particular, DNA and enzyme action (see for instance, [19], [14], and [7]). The structure of the symmetry group has become important information to understand the geometrical, physical properties of knots and links, as well as the enumeration of knots and links (see, for instance, [5] and [6]). As shown in [1], the knots are algebraic, and the symmetry of knots has been one of the interesting topics presented in [18]. In this article, we show heuristically that all composite topological links are actually also algebraic, in particular, the composite link, which is the knot sum of the Hopf link and its mirror image, denoted as 221m!2 2 1, has a symmetry group. The main contribution of this paper is that we show that the symmetry group of the composite link 221m!2 2 1 is (1.1) Z2 × Z2 × Z2 = {1, α} × {1, β} × {1, γ} , where (1.2) α = (1, 1, −1, −1, (2, 3)) , (1.3) β = (1, −1, −1, −1, e), and (1.4) γ = (−1, −1, 1, 1, e). The results can be generalized in the study of the symmetry group of any com- posite topological link, and so it can be used for the classification of composite topological links. This paper is structured as follows: in Section 2, we show the classification of symmetries of the link 221m!2 2 1; and in Section 3, we analyze the structure of the symmetry group from the perspective of algebraic group and prove our main theorem. 2. Classifications Let us compute the compatible (p, r) permutations first. Since the compati- ble permutations just determine which component of each link is connected, the compatible permutations are the same with the case 221!2 2 1. So the compatible permutations are p = (2, 3), r = (1, 2) and p = e, r = e. Now, let’s take the next step, by which we can find p̄1 and p̄2, and indeed, p̄1 = p̄2 = e. Knowing the fact that 221m and 2 2 1 are in different cosets and that 2 2 1 has the symmetry group (2.1) 〈 (1, −1, −1, e), (−1, 1, −1, e), (1, 1, 1, (1, 2) 〉 , we can see that there are 16 cases, as follows, to to considered: (1) γ = (1, 1, 1, 1, (2, 3)); γ1 = (1, 1, 1, e); γ2 = (1, 1, 1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 172 Structure of symmetry group of some composite links and some applications (2) γ = (1, 1, 1, −1, (2, 3)); γ1 = (1, 1, 1, e); γ2 = (1, 1, −1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (3) γ = (1, 1, −1, 1, (2, 3)); γ1 = (1, 1, −1, e); γ2 = (1, 1, 1, e). Since L γ1 2 is in the coset of L1, but L γ2 1 is not in the coset of L2, this case is not in the symmetry group of 221m!2 2 1. (4) γ = (1, 1, −1, −1, (2, 3)); γ1 = (1, 1, −1, e); γ2 = (1, 1, −1, e). Since L γ1 2 is in the coset of L1, and L γ2 1 is in the coset of L2, this case is in the symmetry group of 221m!2 2 1. (5) γ = (1, −1, 1, 1, (2, 3)); γ1 = (1, −1, 1, e); γ2 = (1, −1, 1, e). Since L γ1 2 is in the coset of L1, and L γ2 1 is in the coset of L2, this case is in the symmetry group of 221m!2 2 1. (6) γ = (1, −1, 1, −1, (2, 3)); γ1 = (1, −1, 1, e); γ2 = (1, −1, −1, e). Since L γ2 1 is not in the coset of L2, this case is not in the symmetry group of 221m!2 2 1. (7) γ = (1, −1, −1, 1, (2, 3)); γ1 = (1, −1, −1, e); γ2 = (1, −1, 1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (8) γ = (1, −1, −1, −1, (2, 3)); γ1 = (1, −1, −1, e); γ2 = (1, −1, −1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. The next 8 cases of mirror image are the followings: (9) γ = (−1, 1, 1, 1, (2, 3)); γ1 = (−1, 1, 1, e); γ2 = (−1, 1, 1, e). Since L γ1 2 is in the coset of L1, and and L γ2 1 is in the coset of L2, this case is in the symmetry group of 221m!2 2 1. (10) γ = (−1, 1, 1, −1, (2, 3)); γ1 = (−1, 1, 1, e); γ2 = (−1, 1, −1, e). Since L γ2 1 is not in the coset of L2, this case is not in the symmetry group of 221m!2 2 1. (11) γ = (−1, 1, −1, 1, (2, 3)); γ1 = (−1, 1, −1, e); γ2 = (−1, 1, 1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (12) γ = (−1, 1, −1, −1, (2, 3)); γ1 = (−1, 1, −1, e); γ2 = (−1, 1, −1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (13) γ = (−1, −1, 1, 1, (2, 3)); γ1 = (−1, −1, 1, e); γ2 = (−1, −1, 1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (14) γ = (−1, −1, 1, −1, (2, 3)); γ1 = (−1, −1, 1, e); γ2 = (−1, −1, −1, e). Since L γ1 2 is not in the coset of L1, this case is not in the symmetry group of 221m!2 2 1. (15) γ = (−1, −1, −1, 1, (2, 3)); γ1 = (−1, −1, −1, e); γ2 = (−1, −1, 1, e). Since L γ2 1 is not in the coset of L2, this case is not in the symmetry group of 221m!2 2 1. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 173 Y. Liu (16) γ = (−1, −1, −1, −1, (2, 3)); γ1 = (−1, −1, −1, e); γ2 = (−1, −1, −1, e). Since L γ1 2 is in the coset of L1, and and L γ2 1 is in the coset of L2, this case is in the symmetry group of 221m!2 2 1. In summary, the set of elements involving (2, 3) as the permutation in the symmetry group is (2.2) S1 = {(1, 1, −1, −1, (2, 3)), (1, −1, 1, 1, (2, 3)), (−1, 1, 1, 1, (2, 3)), (−1, −1, −1, −1, (2, 3))} . For the other compatible permutation p = r = e, we can compare L γ1 1 and L1, as well as L γ2 2 and L2 , then we obtain the following set of elements in the symmetry group, (2.3) S2 = {(1, 1, 1, 1, e), (1, −1, −1, −1, e), (−1, −1, 1, 1, e), (−1, 1, −1, −1, e)} . 3. Structure Analysis and Theorem In this section, we analyze the structure of the symmetry group with the multiplication operation of the group and have the following theorem. Theorem 3.1. The symmetry group of the composite link 221m!2 2 1 is isomorphic to Z2 × Z2 × Z2. Proof. Let α = (1, 1, −1, −1, (2, 3)), β = (1, −1, −1, −1, e), γ = (−1, −1, 1, 1, e), δ = (−1, 1, −1, −1, e), and the unit element 1 = (1, 1, 1, 1, e), we have (3.1) (1, −1, 1, 1, (2, 3)) = αβ, (3.2) (−1, 1, 1, 1, (2, 3)) = αδ, and (3.3) (−1, −1, −1, −1, (2, 3)) = αγ. Noticing that δ = βγ, we now have the symmetry group (3.4) G = 〈 1, α, β, γ 〉 . Since any other element in G than the identity is of order 2, then we know that G is abelian. Therefore, G is an abelian group of order 8. By the fundamental theorem of finitely generated abelian group (see for instance [8]), the structure of G is Z8, Z2 × Z4, or Z2 × Z2 × Z2. But since any other element in G than the identity has an order 2, the structure of G must be Z2 × Z2 × Z2. Hence, (3.5) G = {1, α} × {1, β} × {1, γ} ∼= Z2 × Z2 × Z2 where α = (1, 1, −1, −1, (2, 3), β = (1, −1, −1, −1, e), and γ = (−1, −1, 1, 1, e). □ So it turns out that the structure of the symmetry group of the composite link 221m!2 2 1 is the same as the composite link 2 2 1!2 2 1, but the symmetry group of 221m!2 2 1has different elements in its symmetry group. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 174 Structure of symmetry group of some composite links and some applications Remark 3.2. Topological knots and links were studied by using integral ge- ometry, on which one can refer to [13], but other theoretic work of integral geometry such as [2], [16], [4], [15], and [17], may also be used to study knots and links. On the other hand, the fundamental theorem of finitely generated abelian group, in the case if the group is abelian, the classification of finite simple groups (see for instance [11]), and other algebraic theories, can be used to determine the structure of the symmetry group. Another remark about the applications of the symmetry groups of knots we would like to make is Remark 3.3. In some physical movements or processes of DNA, the group structure of the double helix strands of DNA is invariant, and therefore, it can be used to track these movements or processes. Furthermore, the chirality of synthetics molecules (see for instance, [20]), which should be induced by the symmetry groups, can be used to identify synthetics molecules and can be potentially applied to the testing on virus infections, which might potentially help with the control on diseases, in particular, the infectious disease, COVID- 19, in the recent pandemic. Figure 1. Discovered DNA Knot (c.f. 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