Chirality of torus-covering $T^2$-links of degree three

TL;DR

This paper studies the chirality of degree three torus-covering $T^2$-links, analyzing how invariants like triple linking numbers, Fox colorings, and quandle cocycle invariants detect chirality.

Contribution

It determines the quandle cocycle invariant for degree three torus-covering $T^2$-links related to tri-colorings, advancing understanding of their chirality detection.

Findings

Chirality detection varies with different invariants.

The quandle cocycle invariant is explicitly computed for tri-colorings.

Invariants like triple linking numbers and Fox colorings provide partial chirality information.

Abstract

A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair $(a,b)$ of 3-braids satisfying $ab=ba$, denoted by $\mathcal{S}_3(a,b)$. We investigate to what extent the chirality of $\mathcal{S}_3(a,b)$ is detected by invariants such as the triple linking numbers, the number of Fox $p$-colorings, and the quandle cocycle invariant associated with $p$-colorings. In particular, we determine the quandle cocycle invariant for $\mathcal{S}_3(a,b)$ associated with tri-colorings.