Virtual link groups

Bardakov, Valeriy G.; Mikhalchishina, Yu. A.; Нещадим, М. В.

doi:10.1134/s0037446617050032

Cited by 20 publications

(19 citation statements)

References 7 publications

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“…., c n−1 , f m , g m,3 , m ∈ Z. Also, as in the case of the group W B ′ 4 , we can express the generators f m , g m, 3 , m ∈ Z, as words in the generators c 3 , f 0 , f 1 , f 2 . For n ≥ 5, note that W B ′ n is perfect as a quotient of the perfect group V B ′ n .…”

Section: 3mentioning

confidence: 99%

“…and is identity since f 3 m = 1. 5) The relation g m+1,3 a −1 m g m,3 = 1 after substitution g m+1, 3…”

Section: The Commutator Subgroupmentioning

confidence: 99%

“…In particular, like closures of classical braids represent classical knots and links, the closure of virtual braids represent the virtual knots and links (see [9], [10]). On connections of virtual braids with the virtual knot theory, see [3,4]. For a structure of the virtual braid groups, see [2].…”

Section: Introductionmentioning

confidence: 99%

“…b m,1 a m+1 f m+2,1 b −1 m,0 = 1, b m,0 f m+2,0 a −1 m+1 b −1 m,1 = 1, f m,0 c 3 b m+1,1 g m+2,3 b −1 m+1,0 c −1 3 = 1, f m,1 c 3 f m+1,0 a m+1 g m+2,3 b −1 m+1,1 c −1 3 = 1. g m,i c i+1 c i g m+2,i+1 c −1 i c −1 i+1 = 1, g m,i c i+1 a −1 m c i a m+1 g m+2,i+1 a −1 m+1 c −1 i a m c −1 i+1 = 1. Similar to the group W B ′ 4, the firs for relations are equivalent to the relations g m+2,3 = f −1 m+2 f m+1 c −1 3 f m c 3 f −1 m+1 , a m = f m f m+1 . Hence, the additional relations of W B ′ n , ≥ 5 have the form g m+2,3 = f −1 m+2 f m+1 c −1 3 f m c 3 f −1 m+1 , a m = f m f m+1 , g m,i c i+1 c i g m+2,i+1 c −1 i c −1 i+1 = 1,…”

mentioning

confidence: 99%

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Commutator subgroups of virtual and welded braid groups

Bardakov

Gongopadhyay

Нещадим

2019

Int. J. Algebra Comput.

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Let V Bn, resp. W Bn denote the virtual, resp. welded, braid group on n strands. We study their commutator subgroups V B ′ n = [V Bn, V Bn] and, W B ′ n = [W Bn, W Bn] respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that V B ′ n is finitely generated if and only if n ≥ 4, and W B ′ n is finitely generated for n ≥ 3. Also we prove that, and for n ≥ 5 the commutator subgroups V B ′ n and W B ′ n are perfect, i. e. the commutator subgroup is equal to the second commutator subgroup.

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Section: 3mentioning

confidence: 99%

“…and is identity since f 3 m = 1. 5) The relation g m+1,3 a −1 m g m,3 = 1 after substitution g m+1, 3…”

Section: The Commutator Subgroupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Commutator subgroups of virtual and welded braid groups

Bardakov

Gongopadhyay

Нещадим

2019

Int. J. Algebra Comput.

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“…Unlike classical knots virtual ones are not topological objects therefore there is no natural definition of a virtual knot group. There exist several approaches to the definition (see [11,16,1,9,2,8,4,5,13]). Some of the approaches use a knot diagram, other ones use representations of the virtual braid group by automorphisms of some groups.…”

Section: Introductionmentioning

confidence: 99%

Groups of the virtual trefoil and Kishino knots

Bardakov

Mikhalchishina

Нещадим

2018

J. Knot Theory Ramifications

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In the paper [13] for an arbitrary virtual link L three groups G 1,r (L), r > 0, G 2 (L) and G 3 (L) were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also we prove that G 3 distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.In the present work the fact that groups of the virtual trefoil T v G 1,r (T v ) and G 2 (T v ) are not isomorphic to each other will be proven.Note that the most of virtual knot groups do not distinguish the group of the Kishino knot from the trivial one. The group defined in [9] is an exception. In the paper [13] the fact that groups constructed using representations W 1,r and W 2 do not distinguish the Kishino knot from the trivial one is stated. And the natural question whether the group constructed using the representation W 3 distinguishes the Kishino knot from the trivial one was formulated in [13].In present work the positive answer on the question is given.As it is well known if G is a group of a classical knot, then its commutator subgroup coincides with the third term of lower central series. Therefore, the factorization by terms of the lower central series cannot be used for distinguish of knots. On the other hand as it was pointed out in the paper [3] some of virtual link groups are residually nilpotent groups. Recall that a group G is referred to as residually nilpotent if for every nontrivial element g from G there exists a homomorphism ϕ : G −→ N on a nilpotent group N such that ϕ(g) = 1. It is easy to observe that G is residually nilpotent if and only if the intersection of all terms of its lower central series is a trivial group.In the paper [6] was noted that the virtual trefoil group from [1] has different first five terms of lower central series. This fact allows to construct new invariants of virtual links. The question occurs whether virtual link groups are residually nilpotent or not. Moreover in [6] representations of some virtual knot groups into finite algebras were constructed. In the present paper this approach is being developed.Let us describe the paper content. In Section 2 we recall representations constructed in [13] of the virtual braid group V B n into the group of automorphisms of free groups which are extensions of the Wada representations of braid group B n . Also we give presentations of three groups G 1,r , G 2 and G 3 of the virtual trefoil. In Section 3 the structure of these groups is found out. In Section 4, using quotients by the second commutator subgroups, we prove that G 1,r is not isomorphic to G 2 ; we study the quotients by the terms of t...

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Virtual Knot Theory and Virtual Knot Cobordism

Kauffman

2019

Knots, Low-Dimensional Topology and Applications

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We generalize virtual knot theory to multi-virtual knot theory where there are a multiplicity of virtual crossings. Each virtual crossing type can detour over the other virtual crossing types, and over the classical or flat classical crossings. New invariants of multivirtual knots and links are introduced and new problems that arise are described. We show how the extensions of the Penrose coloring evaluation for trivalent plane graphs and our generalizations of this to non-planar graphs and arbitrary numbers of colors acts as a motivation for the construction of the multi-virtual theory.

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Virtual link groups

Cited by 20 publications

References 7 publications

Commutator subgroups of virtual and welded braid groups

Commutator subgroups of virtual and welded braid groups

Groups of the virtual trefoil and Kishino knots

Virtual Knot Theory and Virtual Knot Cobordism

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