Twisted Alexander matrices of quandles associated with a certain Alexander pair

TL;DR

This paper investigates $f$-twisted Alexander matrices of specific quandles derived from quandle 2-cocycles, linking their invariants to surface knot invariants and quandle homology.

Contribution

It demonstrates how the 0-th elementary ideal of these matrices relates to Carter-Saito-Satoh's invariant and explores connections with quandle homology groups.

Findings

0-th elementary ideal describes surface knot invariants

Relationship established between $f$-twisted matrices and quandle homology

Analysis of Alexander pair derived from quandle 2-cocycle

Abstract

Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix, which is a quandle version of a twisted Alexander matrix and defined an invariant of finitely presented quandles. In this paper, we study $f$-twisted Alexander matrices of certain quandles with the Alexander pair obtained from a quandle 2-cocycle. We show that the 0-th elementary ideal of $f$-twisted Alexander matrix of the knot quandle of a surface knot with the Alexander pair obtained from a quandle 2-cocycle can be described with the Carter-Saito-Satoh's invariant. We also discuss a relationship between $f$-twisted Alexander matrices of connected quandles with the Alexander pair obtained from a quandle 2-cocycle and quandle homology groups.