A hyperbolic perspective on the Dehn surgery characterisation problem
File(s)
Author(s)
Wakelin, Laura
Type
Thesis or dissertation
Abstract
As is the case for many questions in low-dimensional topology, the Dehn surgery characterisation problem is simple to state but difficult to solve. Given a non-trivial slope p/q, when does the oriented homeomorphism type of the 3-manifold obtained by Dehn surgery of slope p/q on a knot K in the 3-sphere uniquely characterise the knot K? A wide variety of techniques, ranging from Floer homology to more geometric approaches, have been successfully employed to find characterising slopes for certain knots; the construction of non-characterising slopes has also been studied extensively. However, there is currently no universal criterion to determine whether a slope is characterising or non-characterising for any given knot.
This thesis investigates the Dehn surgery characterisation problem in the case where the knot complement has a hyperbolic outermost JSJ piece. The fundamental strategy involves comparing JSJ decompositions before and after Dehn filling: this allows us to reduce the problem at hand to one about hyperbolic geometry. Subsequently, two novel ideas - minimal geodesics (in the more general case) and minimal volumes (in the special case of Whitehead doubles) - can be used to determine explicit conditions which ensure that a slope is characterising. The limitations of this method are also illustrated via a construction of infinite families of pairs of multiclasped Whitehead doubles sharing a non-characterising slope.
This thesis investigates the Dehn surgery characterisation problem in the case where the knot complement has a hyperbolic outermost JSJ piece. The fundamental strategy involves comparing JSJ decompositions before and after Dehn filling: this allows us to reduce the problem at hand to one about hyperbolic geometry. Subsequently, two novel ideas - minimal geodesics (in the more general case) and minimal volumes (in the special case of Whitehead doubles) - can be used to determine explicit conditions which ensure that a slope is characterising. The limitations of this method are also illustrated via a construction of infinite families of pairs of multiclasped Whitehead doubles sharing a non-characterising slope.
Version
Open Access
Date Issued
2023-08
Online Publication Date
2023-11-20T15:37:44Z
Date Awarded
2023-10
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Sivek, Steven
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/S021590/1
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)