Erratum: Biquandles of small size and some invariants of virtual and welded knots

Fenn, Roger; Bartholomew, Andrew

doi:10.1142/s021821651792002x

Cited by 5 publications

(2 citation statements)

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“…Example 2.5. Let G be a group, and x * y = y −1 x −1 y, x * y = y −2 x for x, y ∈ G. Then (G, * , * ) is a biquandle which is called the Wada biquandle (see [49,54]).…”

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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“…Example 2.5. Let G be a group, and x * y = y −1 x −1 y, x * y = y −2 x for x, y ∈ G. Then (G, * , * ) is a biquandle which is called the Wada biquandle (see [49,54]).…”

Section: Biracks and Biquandlesmentioning

confidence: 99%

General constructions of biquandles and their symmetries

Bardakov¹,

Nasybullov²,

Singh³

2019

Preprint

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“…Problem 7 is related to the construction of finite biquandles of small cardinality, see for example [10,38]. A different approach to construct all solutions could be based on skew left braces, see [2]; this method requires the classification of finite skew left braces.…”

Section: Introductionmentioning

confidence: 99%

Problems on skew left braces

Vendramin

2018

Preprint

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Braces were introduced by Rump as a generalization of Jacobson radical rings. It turns out that braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to the Yang-Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we collect problems related to (skew) braces and set-theoretic solutions to the Yang-Baxter equation.

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A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation

Rump

2019

Journal of Algebra

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Erratum: Biquandles of small size and some invariants of virtual and welded knots

Abstract: Some of the results on welded knots in the title paper were incorrect. In this corrigendum, these are corrected and extended. An additional appendix is also included.

Cited by 5 publications

References 4 publications

General constructions of biquandles and their symmetries

General constructions of biquandles and their symmetries

Problems on skew left braces

A covering theory for non-involutive set-theoretic solutions to the Yang–Baxter equation

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