Homology group of branched cyclic covering over a 2-bridge knot of genus two

TL;DR

This paper determines the structure of the first homology group of 2-fold cyclic coverings over genus two 2-bridge knots, extending previous work from genus one cases and expressing results via the Alexander polynomial.

Contribution

It provides an explicit description of the homology group for genus two 2-bridge knots, expanding the understanding of knot invariants in cyclic coverings.

Findings

Explicit structure of H_1 for 2-bridge genus two knots

Extension of Seifert and Fox methods to higher genus

Results expressed in terms of Alexander polynomial

Abstract

The structure of the first homology group of a cyclic covering of a knot is an important invariant well known in the knot theory. In the last century, H. Seifert developed a general approach to compute the homology group of the covering. Based on his ideas R. Fox found explicit form for $H_{1}(M_{n},\mathbb{Z}),$ where $M_{n}$ is an $n$-fold cyclic covering over a knot $K$ admitting genus one Seifert surface. The aim of the present paper is to find the structure of $H_{1}(M_{n},\mathbb{Z})$ for $2$-bridge knots admitting genus two Seifert surface. The result is given explicitly in terms of Alexander polynomial of the knot.