The Gordian distance of handlebody-knots and Alexander biquandle colorings

Murao, Tomo

doi:10.2969/jmsj/77417741

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…If K = Z[t ±1 , s ±1 ], then this biquandle is called the Alexander biquandle and is denoted by A s,t (M ). If K = Z n , M = K is the additive group of K, and t = −1, then this biquandle is called the dihedral biquandle of order n. Alexander biquandles and dihedral biquandles were studied, for example, in [18,30,40,42].…”

Section: Biracks and Biquandlesmentioning

confidence: 99%

General constructions of biquandles and their symmetries

Bardakov¹,

Nasybullov²,

Singh³

2019

Preprint

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Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.

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Section: Biracks and Biquandlesmentioning

confidence: 99%

General constructions of biquandles and their symmetries

Bardakov¹,

Nasybullov²,

Singh³

2019

Preprint

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“…Then X := Z 3 [t ±1 ]/(f (t)) is a Z 8 -family of Alexander biquandles and a field. By [18,Example 7.3], it follows that dim Col X (D n , φ n ) = n as vector spaces over X, and the assignment of elements x 1 , . .…”

Section: By This Proposition We Have #Colmentioning

confidence: 99%

A relationship between multiple conjugation quandle/biquandle colorings

Murao

2018

Preprint

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We define a functor Q from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle X, there is a one-to-one correspondence between the set of X-colorings and that of Q(X)-colorings diagrammatically for any handlebody-link and spatial trivalent graph. In particular, we prove that the set of G-family of Alexander biquandles colorings is isomorphic to that of G-family of Alexander quandles colorings as modules.

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