A Polynomial Invariant of Finite Quandles

Nelson, Sam

doi:10.1142/s0219498808002801

Cited by 18 publications

(23 citation statements)

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“…A two-variable polynomial invariant of finite quandles, denoted qp Q (s, t), was introduced in [11]. This invariant was shown to distinguish all non-Latin quandles of order 5 and lower.…”

Section: Introductionmentioning

confidence: 99%

Generalized Quandle Polynomials

Nelson¹

2011

Can. math. bull.

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“…A two-variable polynomial invariant of finite quandles, denoted qp Q (s, t), was introduced in [11]. This invariant was shown to distinguish all non-Latin quandles of order 5 and lower.…”

Section: Introductionmentioning

confidence: 99%

Generalized Quandle Polynomials

Nelson¹

2011

Can. math. bull.

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“…This example can be generalized. The following definition and proposition can be found in Section 2 of[20].Definition 5.11. Let Q be a finite quandle.…”

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confidence: 99%

Ring theoretic aspects of quandles

2019

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We associate to every quandle X and an associative ring with unity k, a nonassociative ring k[X] following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2transitively on X, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles R n is given. It is also shown that if for two quandles X and Y the inner automorphism groups act 2-transitively and k[X] is isomorphic to k[Y], then the quandles are of the same partition type. However, we provide examples when the quandle rings k[X] and k[Y] are isomorphic, but the quandles X and Y are not isomorphic. These examples answer some open problems in [3].

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“…This polynomial contains information not just about the subbirack Y itself but also about how Y is embedded in X. See [20] for more.…”

Section: Examplementioning

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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A Polynomial Invariant of Finite Quandles

Abstract: We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.

Cited by 18 publications

References 10 publications

Generalized Quandle Polynomials

Generalized Quandle Polynomials

Ring theoretic aspects of quandles

Link invariants from finite biracks

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