Liam Watson

Associate Professor

Research Classification

Research Interests

Low-dimensional topology
Khovanov homology
Heegaard Floer homology

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I am available and interested in collaborations (e.g. clusters, grants).
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Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Length-minimizing closed curves on manifolds with boundary (2022)

The purpose of this thesis is to explore the properties of closed curves which are length-minimizing in a nontrivial homotopy class of a compact Riemannian manifold with boundary. This project was inspired by a question asked by Professor Liam Watson about the existence of length-minimizing curves on a torus with finitely many disks removed, and whether these curves leave the boundary components tangentially. In our research, we extend the question to compact smooth Riemannian manifolds with smooth boundary. We first discuss some preliminary results to determine an appropriate class of curves to minimize over. We then explore the properties of the length and energy functionals, showing that a minimizer of the energy is also a minimizer of the length. We directly minimize the energy functional to show the existence of a length-minimizing curve in any nontrivial homotopy class of a compact Riemannian manifold with boundary. Finally, we address the regularity of length-minimizing curves, showing that they are piecewise geodesics (possibly with infinitely many pieces).

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The Khovanov homotopy type and Conway mutation (2021)

Khovanov homology is a combinatorially-defined invariant of knots and links, with various generalizations to tangles. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov homology can be recovered. This thesis is primarily a ground-up survey of the Khovanov homotopy type; beginning with the Jones polynomial, we weave our way through Khovanov homology and the Khovanov homotopy type for links, before finishing with the construction of the Khovanov homotopy type for tangles. Throughout, we place a special emphasis on Conway mutation, an operation on links which involves replacing a tangle within a link by a related tangle. Despite its non-triviality, Conway mutation is impossible to detect with the Jones polynomial, and difficult to detect with Khovanov homology. The extent to which the Khovanov homotopy type is able to detect mutation is an open question, and the Khovanov homotopy type for tangles seems to be particularly well-suited for investigating this question.

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Thinking of (1,1) knots using elastic bands on peg-boards and combed glazing on the mille-feuille (2020)

Despite the analytic underpinnings of Heegaard Floer theory and its refine- ment to knots, there is an interesting class of knots, the (1, 1) knots, which have the special property that their knot Floer homology can be computed na ̈ıvely, straight from the definition, using only combinatorial techniques. In this thesis, we survey (1,1)-knots, describe their knot Floer homology, and focus in particular on the landscape of the manifolds obtained by Dehn surgery on these knots. More precisely, J. Greene, S. Lewallen and F. Vafaee recently described a simple criterion for determining if a (1, 1) knot admits a nontrivial surgery to an L-space, using the orientation of the curves in a doubly pointed genus-1 Heegaard diagram for the knot. This character- ization is formally very similar to a characterization due to J. Hanselman, J. Rasmussen and L. Watson, using a graphical calculus they developed for working with the bordered Floer theory. We relate these two perspectives, by providing in the final chapter a novel proof of Greene et al.’s criterion using the graphical calculus, recently expanded by A. Kotelskiy, Watson and C. Zibrowius.

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