Coloring Link Diagrams by Alexander Quandles

Bae, Yongju

doi:10.1142/s0218216512500940

Cited by 13 publications

(24 citation statements)

References 9 publications

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“…The latter is known as Stein's theorem [43] and originally required a rather involved topological argument. Next, we prove that knots and links with trivial Alexander polynomial are not colorable by any solvable quandle (Theorem 8.4), extending an analogical result for affine quandles [3,28]. Finally, we explain the Belousov-Onoi correspondence that translates our results for latin quandles into their loop isotopes.…”

mentioning

confidence: 55%

Commutator theory for racks and quandles

Bonatto¹,

Stanovský²

2019

Preprint

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We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as abelianness and centrality are reflected by the corresponding relative displacement groups, and so do the global properties, solvability and nilpotence. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order 2 k , no latin quandles of order ≡ 2 (mod 4)), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

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mentioning

confidence: 55%

Commutator theory for racks and quandles

Bonatto¹,

Stanovský²

2019

Preprint

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“…Proof of Proposition 1.1 (1). If L is (Z n , ℓ * k )-colorable, then L is (Z p , ℓ * k )-colorable for some odd prime divisor p of n by Remark 1.2 (c).…”

Section: Fundamental Propertiesmentioning

confidence: 91%

“…It is well-known that if the determinant of a given knot is divisible by an integer n with n ≥ 3, then any diagram of the knot is Fox n-colorable [3]. Such colorings are generalized to quandle colorings (see Section 2), and some conditions on the Alexander polynomial of a given knot so that its diagrams have non-trivial colorings with a given Alexander quandle are shown in [1,4].…”

Section: Introductionmentioning

confidence: 99%

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ON LINEAR n-COLORINGS FOR KNOTS

Hayashi

Oshiro

2012

J. Knot Theory Ramifications

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“…The structure of this article is as follows. In section 2 we recall the basic notions of quandles ( [12], [18]), in particular linear Alexander quandles, colourings of knot diagrams ( [2], [4], [10], [11], [17], [19]), the colouring matrix and the Alexander polynomial ( [1], [16]). Section 3 presents the two types of triangularized matrices found when considering colouring matrices for prime knots up to ten crossings and section 4 compares these matrices with those in [15].…”

Section: Introductionmentioning

confidence: 99%

Colourings and the Alexander Polynomial

Camacho¹,

Dionisio²,

Picken³

2016

Kyungpook mathematical journal

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Abstract. Using a combination of calculational and theoretical approaches, we establish results that relate two knot invariants, the Alexander polynomial, and the number of quandle colourings using any finite linear Alexander quandle. Given such a quandle, specified by two coprime integers n and m, the number of colourings of a knot diagram is given by counting the solutions of a matrix equation of the form AX = 0 mod n, where A is the m-dependent colouring matrix. We devised an algorithm to reduce A to echelon form, and applied this to the colouring matrices for all prime knots with up to 10 crossings, finding just three distinct reduced types. For two of these types, both upper triangular, we found general formulae for the number of colourings. This enables us to prove that in some cases the number of such quandle colourings cannot distinguish knots with the same Alexander polynomial, whilst in other cases knots with the same Alexander polynomial can be distinguished by colourings with a specific quandle. When two knots have different Alexander polynomials, and their reduced colouring matrices are upper triangular, we find a specific quandle for which we prove that it distinguishes them by colourings.

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Coloring Link Diagrams by Alexander Quandles

Cited by 13 publications

References 9 publications

Commutator theory for racks and quandles

Commutator theory for racks and quandles

ON LINEAR n-COLORINGS FOR KNOTS

Colourings and the Alexander Polynomial

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