Computation of quandle 2-cocycle knot invariants without explicit 2-cocycles

TL;DR

This paper introduces a method to compute quandle 2-cocycle knot invariants without explicitly finding 2-cocycles, using abelian extensions and generalized Alexander quandles, enabling effective knot distinction.

Contribution

It presents a novel approach to compute 2-cocycle invariants via abelian extensions, simplifying calculations and expanding applicability without explicit cocycle construction.

Findings

Successfully distinguishes all prime knots up to 11 crossings

Most 12-crossing prime knots classified by symmetry

Invariant equivalent to Eisermann's coloring polynomial for certain quandles

Abstract

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding $2$-cocycles. This permits the construction of many $2$-cocycle invariants without exhibiting explicit $2$-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann's knot coloring polynomial. Computations using this technique show that the $2$-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crosssings including classification by symmetry: mirror images, reversals, and reversed mirrors.