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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

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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 (PhD)
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:
DateEvent
August 1982Submitted
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Sponsors: Science Research Council (Great Britain)
Format of File: pdf
Extent: 116 leaves : illustrations
Language: eng

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