Alexander matrices of link quandles associated to quandle homomorphisms and quandle cocycle invariants

TL;DR

This paper explores the relationship between Alexander matrices of link quandles and quandle cocycle invariants, demonstrating that quandle-based invariants can be stronger than traditional group-based invariants for knots.

Contribution

It establishes a connection between $f$-twisted Alexander matrices and quandle cocycle invariants, showing the enhanced strength of quandle invariants over group invariants.

Findings

$f$-twisted Alexander invariant is a stronger oriented knot invariant

The relationship between Alexander matrices and cocycle invariants is clarified

Quandle invariants can detect more knot properties than group invariants

Abstract

A. Ishii and K. Oshiro introduced the notion of an $f$-twisted Alexander matrix. This notion is a quandle version of the twisted Alexander matrix which was introduced by M. Wada. They showed that the twisted Alexander matrix of a pair of a link group and a group representation can be recovered from the $f$-twisted Alexander matrix of a pair of a link quandle and a quandle homomorphism. In this paper, we study a relationship between the $f$-twisted Alexander matrix and the quandle cocycle invariant. As an application, we show that an $f$-twisted Alexander invariant of knot quandles is a really stronger oriented knot invariant than a twisted Alexander invariant of knot groups.