Homological invariants of strongly invertible knots

Snape, Michael (2018) Homological invariants of strongly invertible knots. PhD thesis, University of Glasgow.

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Printed Thesis Information: https://eleanor.lib.gla.ac.uk/record=b3330822

Abstract

This thesis explores the relationship between Khovanov homology and strongly invertible knots through the use of a geometric construction due to Sakuma. On the one hand, new homological and polynomial invariants of strongly invertible knots are extracted from Sakuma's construction, all of which are related to Khovanov homology. Conversely, these invariants are used to study the two-component links and annular knots obtained from Sakuma's construction, the latter of which are almost entirely disjoint from the class of braid closures. Applications include the problem of unknot detection in the strongly invertible setting, the efficiency of an invariant when compared with the $\eta$-polynomial of Kojima and Yamasaki, and the use of polynomial invariants to bound the size of the intrinsic symmetry group of a two-component Sakuma link. We also define a new quantity, $\varkappa_A$, and conjecture that it is an invariant of strongly invertible knots.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Keywords: Knot theory, low-dimensional topology.
Subjects: Q Science > QA Mathematics
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
Funder's Name: Engineering and Physical Sciences Research Council (EPSRC), Engineering and Physical Sciences Research Council (EPSRC), Engineering and Physical Sciences Research Council (EPSRC), Engineering and Physical Sciences Research Council (EPSRC)
Supervisor's Name: Watson, Professor Liam
Date of Award: 2018
Depositing User: Mr Michael Snape
Unique ID: glathesis:2018-39015
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 18 Dec 2018 16:27
Last Modified: 03 Jan 2019 11:22
URI: http://theses.gla.ac.uk/id/eprint/39015

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