Liam Watson

Associate Professor

Research Classification

Research Interests

Low-dimensional topology
Khovanov homology
Heegaard Floer homology

Relevant Degree Programs

Research Options

I am available and interested in collaborations (e.g. clusters, grants).
I am interested in and conduct interdisciplinary research.
I am interested in working with undergraduate students on research projects.


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Graduate Student Supervision

Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

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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Program Affiliations


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