Quandle coloring and cocycle invariants of composite knots and abelian extensions

TL;DR

This paper explores how quandle colorings and cocycle invariants can distinguish composite knots and their properties, providing formulas, computational results, and insights into abelian extensions of quandles.

Contribution

It introduces new methods for computing cocycle invariants of composite knots and examines their relation to abelian quandle extensions, supported by computational data.

Findings

Quandle colorings distinguish certain composite knots like square and granny knots.

Formulas are provided for calculating cocycle invariants from coloring counts.

Computational analysis of small connected quandles (Rig quandles) is presented.

Abstract

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.