Homology of branched cyclic covers of knots
HTML articles powered by AMS MathViewer
- by Stanley Ocken PDF
- Proc. Amer. Math. Soc. 110 (1990), 1063-1067 Request permission
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
- 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
- C. McA. Gordon, Some aspects of classical knot theory, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 1–60. MR 521730 H. Seifert, Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen, Abh. Math. Sem. Univ. Hamburg 11 (1935), 84-101.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1063-1067
- MSC: Primary 57M12; Secondary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-1990-0984809-5
- MathSciNet review: 984809