Detecting non-admissibility of quandles via colorings

TL;DR

This paper introduces criteria to identify non-admissible quandles using colorings of tangles, enabling the construction of new knot invariants beyond traditional group-based methods.

Contribution

It provides a novel method to determine quandle admissibility through tangle colorings and constructs numerous non-admissible quandles from simple knots.

Findings

Criteria established for quandle admissibility via (1, 1)-tangle colorings

Constructed many non-admissible quandles from Hopf link and trefoil knot

Non-admissible quandles can produce new knot invariants

Abstract

A quandle is an algebraic system whose axioms are motivated by Reidemeister moves in knot theory. A typical example is a conjugation quandle arising from a group. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. Admissible quandles often yield knot invariants that coincide with those derived from the knot group, whereas nonadmissible quandles may produce genuinely new invariants. In this sense, it is important to construct non-admissible quandles. In this paper, we provide criteria for determining whether given quandles are admissible by colorings of (1, 1)-tangles. As an application, we construct numerous examples of non-admissible quandles by analyzing simple tangles obtained from the Hopf link and the trefoil knot.