Detecting non-admissibility of quandles via colorings

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.

Figures (5)

• Figure 1: A diagram $\tilde{D}$ of the $(1,1)$-tangle $\tilde{L}$ obtained from $D$
• Figure 2: A coloring condition
• Figure 3: An $X$-colored $(1,1)$-tangle diagram $D$ obtained from a diagram of the Hopf link $2_{1}^{2}$
• Figure 4: An $X$-colored diagram $D$ of the $(1,1)$-tangle obtained from $3_{1}$
• Figure 5: Arcs around the crossing $c$

Theorems & Definitions (34)

• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Proposition 3.4