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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Homology of branched cyclic covers of knots

Author: Stanley Ocken
Journal: Proc. Amer. Math. Soc. 110 (1990), 1063-1067
MSC: Primary 57M12; Secondary 57M25
MathSciNet review: 984809
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Abstract: This paper presents a new formula for the first integral homology group of the branched cyclic $ p$-fold cover $ {\Sigma _p}$ of a knot $ K$ in the $ 3$-sphere. Given a diagram of $ K$ with $ k$ crossings, let $ A(t)$ be the $ (k - 1) \times (k - 1)$ Alexander matrix of the diagram. Let $ C = A{(1)^{ - 1}}A(0)$, and let $ I$ be the identity matrix. Then $ {(C - I)^p} - {C^p}$ is a presentation matrix for $ {H_1}({\Sigma _p})$.

References [Enhancements On Off] (What's this?)

  • [F] R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
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Keywords: Branched cyclic coverings
Article copyright: © Copyright 1990 American Mathematical Society

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