Classification of fused links

Nasybullov, Timur

doi:10.1142/s0218216516500760

Cited by 15 publications

(14 citation statements)

References 20 publications

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“…Remark 1.4. Theorem 1.3 easily follows from the classification of welded links up to UC-moves, given in [24,Theorem 8] and [22,Theorem 4.7]. In this paper, we will prove Theorem 1.3 without using their classification.…”

Section: (N)mentioning

confidence: 89%

Generalized virtualization on welded links

Miyazawa¹,

Wada

Yasuhara³

2020

J. Math. Soc. Japan

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Let n be a positive integer. The aim of this paper is to study two local moves V (n) and V n on welded links, which are generalizations of the crossing virtualization. We show that the V (n)-move is an unknotting operation on welded knots for any n, and give a classification of welded links up to V (n)-moves. On the other hand, we give a necessary condition for which two welded links are equivalent up to V n -moves. This leads to show that the V n -move is not an unknotting operation on welded knots except for n = 1. We also discuss relations among V n -moves, associated core groups and the multiplexing of crossings.

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Section: (N)mentioning

confidence: 89%

Generalized virtualization on welded links

Miyazawa¹,

Wada

Yasuhara³

2020

J. Math. Soc. Japan

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“…Generally, the µ-component virtual links L = K 1 ∪• • •∪K µ up to forbidden moves are classified by the (i, j)-linking numbers Lk(K i , K j ) ∈ Z (1 ≤ i = j ≤ µ) completely (cf. [1,7,10]). A CF-move introduced by Oikawa in [9] is a local move on virtual link diagrams as shown in Figure 1.4.…”

Section: 1 Two Types Of Double Points On Virtual Link Diagramsmentioning

confidence: 99%

CF-moves for virtual links

Wada

2021

Preprint

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Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CFmoves by the virtual linking number and his n-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawa's n-invariant and introduce two invariants for 3-component even virtual links. Using these invariants together with the virtual linking number, we classify 3-component even virtual links up to CF-moves. As a result, a classification of 3-component virtual links up to CF-moves is provided.

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“…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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Classification of fused links

Cited by 15 publications

References 20 publications

Generalized virtualization on welded links

Generalized virtualization on welded links

CF-moves for virtual links

A journey through loop braid groups

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