Algebraic Properties of Quandle Extensions and Values of Cocycle Knot Invariants

TL;DR

This paper investigates algebraic properties of quandle extensions and their impact on cocycle knot invariants, revealing conditions under which these invariants are constant or restricted for classical knots.

Contribution

It establishes new algebraic conditions that determine when quandle cocycle invariants are constant or restricted, especially for abelian extensions that are conjugation quandles.

Findings

Cocycle invariants are constant under certain algebraic conditions.

Abelian extensions as conjugation quandles yield constant invariants.

Examples from small connected quandles illustrate the theoretical results.

Abstract

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.