• Kyungpook Mathematical Journal
• /
• v.63 no.4
• /
• pp.709-718
• /
• 2023

The inner automorphism groups of quandles are related to the classification problem of quandles. The inner automorphism group of a quandle is generated by inner automorphisms which are presented by columns in the operation table of the quandle. In this paper, we describe inner automorphisms of an abelian extension of a quandle by expressing columns of the operation table of the extended quandle as columns of the operation table of the original quandle. Such a description will be helpful in studying inner automorphism groups of abelian extensions of quandles.

• Kyungpook Mathematical Journal
• /
• v.64 no.2
• /
• pp.311-335
• /
• 2024

The fundamental quandle is a powerful invariant of knots, links and spatial graphs, but it is often difficult to determine whether two quandles are isomorphic. One approach is to look at quotients of the quandle, such as the n-quandle defined by Joyce [8]; in particular, Hoste and Shanahan [5] classified the knots and links with finite n-quandles. Mellor and Smith [12] introduced the N-quandle of a link as a generalization of Joyce's n-quandle, and proposed a classification of the links with finite N-quandles. We generalize the N-quandle to spatial graphs, and investigate which spatial graphs have finite N-quandles. We prove basic results about N-quandles for spatial graphs, and conjecture a classification of spatial graphs with finite N-quandles, extending the conjecture for links in [12]. We verify the conjecture in several cases, and also present a possible counterexample.

• Kyungpook Mathematical Journal
• /
• v.60 no.2
• /
• pp.279-288
• /
• 2020

Niebrzydowski and Przytycki conjectured that the torsion of rack and quandle homology of a dihedral quandle of order 2k is annihilated by k, unless k = 2^t for t > 1. We partially prove this conjecture.

• Kyungpook Mathematical Journal
• /
• v.57 no.4
• /
• pp.711-719
• /
• 2017

Let Inn(Q) denote the inner automorphism group on a quandle Q. For a subset M of Q, let c(M) denote the orbit of M under the Inn(Q)-action on Q. Then c satisfies the axioms of the closure operator. In this paper, we study the topological space Q corresponding to the topology obtained from the closure operator c.

• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.1017-1045
• /
• 2016

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.

• Kyungpook Mathematical Journal
• /
• v.56 no.1
• /
• pp.1-27
• /
• 2016

We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from 3-Lie algebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We introduce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.

• East Asian mathematical journal
• /
• v.30 no.1
• /
• pp.51-62
• /
• 2014

The purpose of this paper is to explicitly present Andruskiewitsch-Gra$\tilde{n}$a's dynamical cocycles associated with the two known non-medial left-distributive quasigroups of order 15 which are extensions of the dihedral quandle of order 3 by those cocycles.