Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T01:54:59.573Z Has data issue: false hasContentIssue false
  • English
  • Français

Homology Invariants

Published online by Cambridge University Press:  20 November 2018

Richard Hartley  and
Kunio Murasugi
Richard Hartley
Affiliation:
University of Toronto, Toronto, Ontario
Kunio Murasugi
Affiliation:
University of Toronto, Toronto, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 655 - 670
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Brody, E. J., The topological classification of the lens spaces, Annals of Mathematics 71 (1960), 163–184.Google Scholar
2. Burde, G., On branched coverings of S3, Can. J. Math. 23 (1971), 84–89.Google Scholar
3. Burnside, W., Theory of groups of finite order (Dover Publications, Inc. Second edition, 1911).Google Scholar
4. Fox, R. H., A quick trip through knot theory, Topology of three manifolds and related topics Prentice Hall, 1962), 120–167.Google Scholar
5. Hartley, R. and Murasugi, K., Covering linkage invariants, Can. J. Math. 29 (1977), 1312–1339.Google Scholar
6. MacLane, S., Categories for the working mathematician (Springer-Verlag, New York, 1971).Google Scholar
7. Perko, K. A., Jr., Octahedral knot covers, Knots, groups and 3-manifolds (Papers dedicated to the memory of Fox, R. H., ed. L. P. Neuwirth), Annals of Math. Studies 84 (1975), 47–50.Google Scholar
8. Reidemeister, K., Knoten und Verkettungen, Math. Zeit. 29 (1929), 713–729.Google Scholar
9. Riley, R., Homomorphisms of knot groups on finite groups, Mathematics of Computation, 25 (1971), 603–619.Google Scholar
10. Riley, R., Hecke invariants of knot groups, Glasgow Math. J. 15 (1974), 17–26.Google Scholar
11. Spanier, E. H., Algebraic topology (McGraw-Hill Book Company, New York, 1966).Google Scholar