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

2021/11/08 by I. A. Mednykh, Ilya Mednykh, Mednykh, Ilya
Mathematics · Medicine · #39A06 #57K14 #57M12 #Botulinum Toxin and Related Neurological Disorders #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.CO #msc:39A06 #msc:57K14 #msc:57M12

paper · pdf · doi:10.48550/arxiv.2111.04292

arxiv created 2021/11/08 · openalex publication_date 2021/11/08 · arxiv updated 2021/11/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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 H₁(M_n,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 H₁(M_n,ℤ) for 2-bridge knots admitting genus two Seifert surface. The result is given explicitly in terms of Alexander polynomial of the knot.

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