On Homology of Virtual Braids and Burau Representation

Vershinin, V. V.

doi:10.1142/s0218216501001165

Cited by 72 publications

(68 citation statements)

References 5 publications

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“…If we omit relations (2.22) we obtain the "virtual braid group" VB n of Vershinin [39]. This plays a role in virtual knot theory analogous to that of the usual braid group in ordinary knot theory.…”

Section: John C Baez Derek K Wise and Alissa S Cransmentioning

confidence: 99%

Exotic statistics for strings in 4d BF theory

Baez¹,

Wise²,

Crans³

2007

Advances in Theoretical and Mathematical Physics

125

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After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the "loop braid group". This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the "braid permutation group" of Fenn, Rimányi, and Rourke. In the context of 4d BF theory, this group naturally acts on the moduli space of flat G-bundles on the complement of a collection of unlinked unknotted circles in R 3 . When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss "quandle field theory", in which the gauge group G is replaced by a quandle.e-print archive: http://lanl.arXiv.org/abs/arXiv:gr-qc/0603085

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Section: John C Baez Derek K Wise and Alissa S Cransmentioning

confidence: 99%

Exotic statistics for strings in 4d BF theory

Baez¹,

Wise²,

Crans³

2007

Advances in Theoretical and Mathematical Physics

125

View full text Add to dashboard Cite

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“…In [15] S. Kamada proves a Markov Theorem for virtual braids, giving a set of moves on virtual braids that generate the same equivalence classes as the virtual link types of their closures. For reference to previous work on virtual links and braids the reader should consult [2,6,7,9,11,13,14,15,16,18,19,20,21,22,23,27,28,32,36,37].…”

Section: Virtual Braidsmentioning

confidence: 99%

Virtual Braids and the L-Move

Kauffman

Lambropoulou

2006

J. Knot Theory Ramifications

103

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In this paper we prove a Markov Theorem for the virtual braid group and for some analogs of this structure. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is to classical knots and links. In classical knot theory the braid group gives a fundamental algebraic structure associated with knots. The Alexander Theorem tells us that every knot or link can be isotoped to braid form. The capstone of this relationship is the Markov Theorem, giving necessary and sufficient conditions for two braids to close to the same link (where sameness of two links means that they are ambient isotopic).The Markov Theorem in classical knot theory is not easy to prove. The Theorem was originally stated by A.A. Markov with three moves and then N. Weinberg reduced them to the known two moves [29,38]. The first complete proof is due to J. Birman [4]. Other published proofs are due to D. Bennequin [3], H. Morton [30], P. Traczyk [35] and S. Lambropoulou [24,25]. In this paper we shall follow the "L-Move" methods of Lambropoulou. In the L-move approach to the Markov theorem, one gives a very simple uniform move that can be applied anywhere in a braid to produce a braid with the same closure. This move, the L-move, consists in cutting a strand of the braid and taking the top of the cut to the bottom of the braid (entirely above or entirely below the braid) and taking the bottom of the cut to the top of the braid (uniformly above or below in correspondence with the choice for the other end of the cut). See Figure 15 for an illustration of a classical L-move. One then proves that two braids have the same closure if and only if they are related by a sequence of L-moves. Once this L-Move Markov Theorem is established, one can reformulate the result in various ways, including the more algebraic classical Markov Theorem that uses conjugation and stabilization moves to relate braids with equivalent closures.Up to now [25,10,26] the L-moves were only used for proving analogues of the Markov theorem for classical knots and links in 3-manifolds (with or without boundary). Our approach to a Markov Theorem for virtual knots and links follows a similar strategy to the classical case, but necessarily must take into account properties of virtual knots and links that diverge from the classical case. In particlular, we use L-moves that are purely virtual, as well as considering the effect of allowed and forbidden moves of the virtual

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“…The virtual braid group (see [23], [11]) is one of generalizations of the classical braid group. A natural question about the representation of the virtual braid group by automorphisms of some group arises.…”

Section: Introductionmentioning

confidence: 99%

Representations of virtual braids by automorphisms and virtual knot groups

Bardakov

Mikhalchishina

Нещадим

2017

J. Knot Theory Ramifications

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In the present paper the representation of the virtual braid group V B n into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for n ≥ 4 is verified. Using representations of V B n , the virtual link group is defined. Also representations of welded braid group W B n are constructed and the welded link group is defined.

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On Homology of Virtual Braids and Burau Representation

Cited by 72 publications

References 5 publications

Exotic statistics for strings in 4d BF theory

Exotic statistics for strings in 4d BF theory

Virtual Braids and the L-Move

Representations of virtual braids by automorphisms and virtual knot groups

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