Research
I study problems in low-dimensional topology, using invariants like Khovanov homology and Heegaard Floer homology.
This has involved working with bordered Floer homology, towards understanding the link between taut foliations, left-orders, and non-L-spaces.
More recently, my work has focussed on relative versions of these invariants and reinterpreting them in terms of curves in surfaces.
My research is supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC),
and I am privileged to carry out my work on the traditional, ancestral, and unceded territory of the xwmǝθkwǝy̓ǝm (Musqueam) People.
For 2024-2026 I hold a Killam Accelerator Research Fellowship.
I am not really on Instagram @mathonmydeskrightnow all that often, but Mathematical Research Postcards is worth checking out.
My office is in Mathematics 219. The best way to contact me is by email: liam[at]math[dot]ubc[dot]ca
Publications
Preprints and works in preparation
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On mutation invariance in Khovanov homology
(with Artem Kotelskiy and Claudius Zibrowius)
Preprint, arXiv.2602.23442.
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Khovanov multicurves are linear
(with Artem Kotelskiy and Claudius Zibrowius)
Preprint, arXiv.2105.06308.
This paper will be re-worked as a survey article as the results are subsumed by parts of the paper above.
-
Khovanov invariants via Fukaya categories: The tangle invariants agree
(with Artem Kotelskiy and Claudius Zibrowius)
Submitted, arXiv.2004.01619.
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Immersed curves in Khovanov homology
(with Artem Kotelskiy and Claudius Zibrowius)
Submitted, arXiv.1910.14584.
Videos of talks on this material from Institut Fourier (2024):
Part 1,
Part 2,
Part 3,
and
Part 4.
The computational software kht++ associated with these invariants, due to Claudius Zibrowius, is available here.
Published, in press, or accepted
-
Thin links and Conway spheres
(with Artem Kotelskiy and Claudius Zibrowius)
Compositio Mathematica, Volume 160 Number 7 (2024).
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Cosmetic operations and Khovanov multicurves
(with Artem Kotelskiy, Tye Lidman, Allison Moore and Claudius Zibrowius)
Mathematische Annalen, Volume 389 Number 3 (2024).
-
Bordered Floer homology for manifolds with torus boundary via immersed curves
(with Jonathan Hanselman and Jake Rasmussen)
Journal of the American Mathematical Society, Volume 37 Number 2 (2024).
Slides from the Mathematical Congress of the Americas (2017).
Videos of talks on this material from CMO (2017):
Part 1
and
Part 2.
Other talks on this material were recorded at INI (2017) as part of the program Homology theories in low-dimensions.
Videos of talks on (older versions of) this material from BIRS (2016):
Part 1
and
Part 2
-
Heegaard Floer homology for manifolds with torus boundary: properties and examples
(with Jonathan Hanselman and Jake Rasmussen)
Proceedings of the London Mathematical Society, Volume 125 Number 4 (2022), arXiv.1810.10355.
Open access link for this paper.
This includes some older writing on Heegaard Floer homology solid tori which never appeared, though there are videos of talks on this material from BIRS
and from SCGP as well as slides from the AMS Joint Meetings (2013) and from the CMS Winter Meeting (2013).
-
A mnemonic for the Lipshitz–Ozsváth–Thurston correspondence
(with Artem Kotelskiy and Claudius Zibrowius)
Algebraic and Geometric Topology, Volume 23 Number 6 (2023).
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Khovanov homology and strong inversions
(with Artem Kotelskiy and Claudius Zibrowius)
The Open Book Series Volume 5 (2022), arXiv.2104.13592.
Open access link for this paper.
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Cabling in terms of immersed curves
(with Jonathan Hanselman)
Geometry & Topology, Volume 27 Number 3 (2023).
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A calculus for bordered Floer homology
(with Jonathan Hanselman)
Geometry & Topology, Volume 27 Number 3 (2023).
Slides from the EMS/SCM Joint Meetings (2015).
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Symmetry & Mutation, Mathematical Research Postcards, Volume 1 Number 1, 2021.
@mathematicalresearchpostcards
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A refinement of Khovanov homology
(with Andrew Lobb)
Geometry & Topology, Volume 25 Number 4 (2021).
Some additional comments on separating mutants, which simplify/clarify/generalize a construction in this paper, can be found here.
These are incorporated into the published version of the paper.
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L-spaces, taut foliations, and graph manifolds
(with Jonathan Hanselman, Jake Rasmussen and Sarah Rasmussen)
Compositio Mathematica Volume 156 Number 3 (2020), arXiv.1508.05911.
This paper used to be called Taut foliations on graph manifolds.
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On the geography and botany of knot Floer homology
(with Matt Hedden)
Selecta Mathematica, Volume 24 Number 2 (2018).
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Khovanov homology and the symmetry group of a knot
Advances in Mathematics, Volume 313 (2017).
Accompanying Mathematica notebook for some of the calculations for this paper using KnotTheory`.
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The Alexander invariant, Seifert forms, and categorification
(with Jen Hom and Tye Lidman)
Journal of Topology, Volume 10 Number 1 (2017).
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Non-fibered L-space knots
(with Tye Lidman)
Pacific Journal of Mathematics, Volume 267, Number 2 (2014).
- Genus one open books with non-left-orderable fundamental group
(with Yu Li)
Proceedings of the American Mathematical Society, Volume 142 Number 4 (2014).
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Graph manifolds, left-orderability and amalgamation
(with Adam Clay and Tye Lidman)
Algebraic and Geometric Topology, Volume 13, Number 4 (2013).
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On L-spaces and left-orderable fundamental groups
(with Steve Boyer and Cameron Gordon)
Mathematische Annalen, Volume 356, Issue 4 (2013).
- New proofs of certain finite filling results via Khovanov homology
Quantum Topology, Volume 4 Number 4 (2013).
- Turaev torsion, definite 4-manifolds and quasi-alternating knots
(with Josh Greene)
Bulletin of the London Mathematical Society, Volume 45 Number 5 (2013).
- Left-orderable fundamental groups and Dehn surgery
(with Adam Clay)
International Mathematics Research Notices, Number 12 (2013).
- Surgery obstructions from Khovanov homology
Selecta Mathematica Volume 18 Number 2 (2012).
Slides from my talk at the Georgia Topology Conference.
- On cabled knots, Dehn surgery, and left-orderable fundamental groups
(with Adam Clay)
Mathematical Research Letters, Volume 18 Number 6 (2011)
- A surgical perspective on quasi-alternating links
Low-dimensional and Symplectic Topology, Volume 82 of Proceedings of Symposia in Pure Mathematics (2011)
- Does Khovanov homology detect the unknot?
(with Matt Hedden)
American Journal of Mathematics, Volume 132 Number 5 (2010).
- A remark on Khovanov homology and two-fold branched covers
Pacific Journal of Mathematics, Volume 245 Number 2 (2010).
Accompanying Mathematica notebook performing the calculations in this paper using KnotTheory`.
- Knots with identical Khovanov homology
Algebraic and Geometric Topology, Volume 7 (2007).
- Simplification of scalar data via monotone-light factorizations (with Martin Brooks)
Proceedings of the 19th Canadian Conference on Computational Geometry, (2007).
- Any tangle extends to non-mutant knots with the same Jones polynomial
Journal of Knot Theory and its Ramifications, Volume 15 Number 9 (2006).
Edited volumes
- Gauge theory and low-dimensional topology: progress and interactions.
Editors: John Baldwin,
Hans Boden,
John Etnyre,
and Liam Watson
The Open Book Series Volume 5, 2022.
- Characters in low-dimensional topology. A conference celebrating the work of Steven Boyer.
Editors: Olivier Collin,
Cameron Gordon,
Stefan Friedl,
Stephan Tillmann,
and Liam Watson
Contemporary Mathematics 760. Centre de Recherches Mathématiques Proceedings, 2020.
Theses
- Involutions on 3-manifolds and Khovanov homology
UQÀM
PhD thesis (June 2009)
Supervisor: Steve Boyer
- Knots, tangles and braid actions
UBC
MSc thesis (October 2004)
Supervisor: Dale Rolfsen
