Torus-covering knot groups and their irreducible metabelian $SU(2)$-representations

TL;DR

This paper investigates the number of irreducible metabelian $SU(2)$-representations of torus-covering $T^2$-knot groups, relating it to the knot determinant and Fox's $p$-colorability, extending classical knot results.

Contribution

It provides a formula for counting irreducible metabelian $SU(2)$-representations of torus-covering $T^2$-knot groups based on knot determinants.

Findings

Number of irreducible metabelian $SU(2)$-representations is determined by the knot determinant.

The count is related to Fox's $p$-colorability.

Results extend classical knot group representation theory.

Abstract

A torus-covering $T^2$-knot is a surface-knot of genus one determined from a pair of commutative braids. For a torus-covering $T^2$-knot $F$, we determine the number of irreducible metabelian $SU(2)$-representations of the knot group of $F$ in terms of the knot determinant of $F$. It is similar to the result due to Lin for the knot group of a classical knot. Further, we investigate the number of irreducible metabelian $SU(2)$-representations using Fox's $p$-colorability.