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Signatures of surfaces in 3-manifolds and applications to knot and link cobordism
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Cooper, Daryl (1982) Signatures of surfaces in 3-manifolds and applications to knot and link cobordism. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3229974~S15
Abstract
This thesis has two chapters. The first investigates necessary conditions for a classical knot to be slice, improving on some results obtained by Casson and Gordon. The method is to study a Seifert form on an arbitrary surface in an arbitrary 3-manifold M. By analogy to the Seifert form of a knot, certain numerical signature invariants of the surface are defined. These signatures turn out to be bounded when a closed surface bounds a 3-manifold in some 4-manifold whose boundary is M; this is the principle tool. It is used to study surfaces lying in certain cyclic coverings of a knot. A non-embedding result is given for 3-manifolds in 4-manifolds in which β₁ = 0.
Chapter two is an analysis of the Z ⊕ Z cover of a classical link of two components studied by means of a generalisation of the Seifert pairing defined on transverse Seifert surfaces for the link components. This enables a signature function to be defined on the torus generalising the knot signature function on the circle. A new proof is given of the form of the Alexander polynomial for a slice link. A proof of some of Conway's relations between link polynomials is given. And in §4, certain polynomials are shown to arise from links, showing in particular that the Torres conditions are sufficient for linking number two when the components are unknotted.
| Item Type: | Thesis or Dissertation (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Three-manifolds (Topology), Cobordism theory, Knot theory, Link theory | ||||
| Official Date: | August 1982 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Sponsors: | Science Research Council (Great Britain) | ||||
| Format of File: | |||||
| Extent: | 116 leaves : illustrations | ||||
| Language: | eng |
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