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

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.