Biquandles of Small Size and some Invariants of Virtual and Welded Knots

Bartholomew, Andrew; Fenn, Roger

doi:10.48550/arxiv.1001.5127

Cited by 3 publications

(4 citation statements)

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“…One would expect to find similar results to the linear biquandle structures studied here. Additionally, Bartholomew and Fenn [2] point out the nonlinear biquandles of Wada [27] and Silver and Williams [25]. It is hopeful that additional useful examples will arise from these structures.…”

Section: Further Questions and Remarksmentioning

confidence: 99%

Parity Biquandles

Kaestner¹,

Kauffman²

2011

Preprint

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We use crossing parity to construct a generalization of biquandles for virtual knots which we call Parity Biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic Biquandles. For a particular construction named the z-Parity Alexander Biquandle we show that the associated polynomial yields a lower bound on the number of odd crossings as well as the total number of real crossings and virtual crossings for the virtual knot. Moreover we extend this construction to links to obtain a lower bound on the number of crossings between components of a virtual link.

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Section: Further Questions and Remarksmentioning

confidence: 99%

Parity Biquandles

Kaestner¹,

Kauffman²

2011

Preprint

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“…Semiarcs are edges in the graph obtained from a link diagram by replacing crossing points with vertices. Previous work has been done on the semiarc-generated algebraic structures arising from unframed oriented link diagrams, known as biquandles [2,9,15,24]. A special case of biquandles, applicable to flat virtual links or virtual strings and known as semiquandles, is examined in [13] with further examples appearing in [12].…”

Section: Introductionmentioning

confidence: 99%

Link invariants from finite biracks

Nelson

2014

Banach Center Publ.

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A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, (t, s)-racks, Alexander biquandles and Silver-Williams switches, known as (τ, σ, ρ)-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.

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“…Comparing isomorphism classes of fundamental biquandles directly is generally impractical, so for more practical biquandle-derived invariants we can either look to functorial invariants like the Alexander and quaternionic biquandle polynomials studied in [2,3,4,5] which generalize the classical Alexander polynomial, or to representational invariants such as the counting invariant Φ Z X (L) = |Hom(F B(L), X)| where X is a finite biquandle [10,18].…”

Section: Introductionmentioning

confidence: 99%

“…Racks are the objects analogous to quandles which are appropriate for defining representational invariants of blackboard-framed oriented knots and links [12]. Racks are a special case of biracks, recently studied in papers such as [3] and [17].…”

Section: Introductionmentioning

confidence: 99%

Bikei, Involutory Biracks and Unoriented Link Invariants

Aksoy

Nelson

2012

J. Knot Theory Ramifications

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We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N = 1 are biquandles, which we call bikei or 双圭. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.

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Biquandles of Small Size and some Invariants of Virtual and Welded Knots

Cited by 3 publications

References 0 publications

Parity Biquandles

Parity Biquandles

Link invariants from finite biracks

Bikei, Involutory Biracks and Unoriented Link Invariants

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