Unrestricted virtual braids, fused links and other quotients of virtual braid groups

“…In [32] the authors show interest also in certain remarkable quotients of loop braid groups, the symmetric loop braid groups (also known as unrestricted virtual braid groups [37]). These groups have a very simple algebraic structure and we refer to [4,10,42] for their properties and applications to fused links.…”

Section: An Historical Note and Other Referencesmentioning

confidence: 99%

A journey through loop braid groups

Damiani

2017

Expositiones Mathematicae

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Abstract. In this paper we introduce distinct approaches to loop braid groups, a generalization of braid groups, and unify all the definitions that have appeared so far in the literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we introduce an extension of these groups that appears to be a more natural generalization of braid groups from the topological point of view. Throughout the text we motivate the interest in studying loop braid groups and give references to some of their applications.

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“…The group of unrestricted virtual braids, which we will denote throughout this article by U V B n , was introduced by Kauffman and Lambropoulou in [KL04] and [KL06], where they provide a new method for converting virtual knots and links to virtual braids and they prove a Markov Theorem for the virtual braid groups. The group U V B n also appears in [KMRW17] as a quotient of the welded braid group W B n , which is a 3-dimensional analogue of the Artin braid groups B n , and moreover in [BBD15] where Bardakov-Bellingeri-Damiani give a description of the structure of this group.…”

Section: Introductionmentioning

confidence: 99%

“…In [[BBD15], Theorem 2.7], Bardakov-Bellingeri-Damiani gave a presentation of the unrestricted virtual pure braid group, from which one can see that U V P n is isomorphic to the direct product of n(n−1) 2 copies of the free group of rank 2. Thus, it follows that the group U V P n is a right-angled Artin group.…”

Section: Introductionmentioning

confidence: 99%

The unrestricted virtual braid groups $UVB_n$

Makri¹

2021

Preprint

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Let U V Bn and U V Pn be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups U V Bn and U V Pn, and our main results are as follows: for n ≥ 5, we give a complete description, up to conjugation, to all possible homomorphisms from U V Bn to the symmetric group Sn. For n ≥ 3, we characterise all possible images of U V Bn, under a group homomorphism, to any finite group G. For n ≥ 5, we prove that U V Pn is a characteristic subgroup of U V Bn. In addition, we determine the automorphism group of U V Pn and we prove that Z 2 × Z 2 is a subgroup of the outer automorphism group of U V Bn. Lastly, we show that U V Bn and U V Pn are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group W Bn and we discuss about its automorphism group.

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Unrestricted virtual braids, fused links and other quotients of virtual braid groups

Abstract: Abstract. We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.

Cited by 25 publications

References 27 publications

Classification of fused links

Classification of fused links

A journey through loop braid groups

The unrestricted virtual braid groups $UVB_n$

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