University of Glasgow
School of Mathematics and Statistics
My area of research is low-dimensional topology. I am interested in Khovanov homology and Heegaard Floer homology, and my work is informed by connections between these two theories. To date, this has fallen along two distinct but, as it happens, quite interconnected lines of inquiry:
Symmetries and Khovanov homology
The relationship between Khovanov homology and Heegaard Floer homology by way of double branched covers suggests where to look for new applications in Khovanov homology. This seems to strengthen the connection between the two theories and gives rise, for example, to an algebraic tool for studying the mapping class group of the knot complement (also known as the knot's symmetry group).
L-spaces and left-orderability
The simplest three-manifolds, from the vantage point of Heegaard Floer theory, are L-spaces. These come up in application and satisfy interesting properties, but a complete understanding of L-spaces is lacking. However, there appear to be strong connections with foliations and left-orderable groups.
My papers on the arXiv,
These are listed below in (approximate) reverse chronological order.
Preprints and works in preparation
Bordered Floer homology for manifolds with torus boundary via immersed curves
(with Jonathan Hanselman and Jake Rasmussen)
Taut foliations on graph manifolds
(with Jonathan Hanselman, Jake Rasmussen and Sarah Rasmussen)
A calculus for bordered Floer homology (with Jonathan Hanselman)
Slides from the EMS/SCM Joint Meetings (2015)
Khovanov homology and the symmetry group of a knot
Accompanying Mathematica notebook for some of the calculations for this paper using KnotTheory`.
Heegaard Floer homology solid tori (with Jonathan Hanselman)
Videos of talks on this material from BIRS
and from SCGP.
Slides from the AMS Joint Meetings (2013) and from the CMS Winter Meeting (2013).
The Alexander invariant, Seifert forms, and categorification
(with Jen Hom and Tye Lidman)
On the geography and botany of knot Floer homology
(with Matt Hedden)
Published, in press, or accepted
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).
Graph manifolds, left-orderability and amalgamation
(with Adam Clay and Tye Lidman)
Algebraic and Geometric Topology, Volume 13, Number 4 (2013).
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).
- Involutions on 3-manifolds and Khovanov homology
PhD thesis (June 2009)
Supervisor: Steve Boyer
- Knots, tangles and braid actions
MSc thesis (October 2004)
Supervisor: Dale Rolfsen
- Simplification of sampled scalar fields by removal of extrema
(with Martin Brooks)
NRC Technical Report (2007).
- Mel Chen (Glasgow, current).
Lord Kelvin Adam Smith scholarship.
Joint project with Kathryn Elmer in evolutionary biology.
- Michael Snape (Glasgow, current).
- Joseph MacColl (Glasgow, current).
MSci project: Mutation invariance in Khovanov homology after Bloom
- Andrew MacPherson (Glasgow, 2014-2015).
MSci project: Some exact sequences in Khovanov homology
- Joseph MacColl (Glasgow, 2014).
Project: Generalised mutation in Khovanov homology (Carnegie funded)
Paper: Rotors in Khovanov homology, published in Canadian Mathematical Bulletin (online: dx.doi.org/10.4153/CMB-2015-034-6)
Joseph MacColl has completed Level 4 Honours in Glasgow, and is starting his MSci.
- Jeff Hicks (UCLA, 2012).
Project: Modules and filtrations in Khovanov homology (VIGRE funded)
Notes: Module and filtered knot homology theories
Jeff Hicks is now pursuing a PhD at Berkeley.
- Isaac Solomon (UCLA, 2012).
Project: Non-left-orderable Dehn surgeries (VIGRE funded)
Isaac Solomon is now pursuing a PhD at Brown.
- Yu Li (UCLA, 2011).
Project: Orderability and genus one open books (CSST funded)
Paper: Genus one open books with non-left-orderable fundamental group, Proceedings of the American Mathematical Society, Volume 142 Number 4 (2014).
Yu Li is now pursuing a PhD at Wisconsin-Madison.
- Jose Chavez (UCLA, 2011).
Project: The Jones polynomial and Khovanov homology
Jose Chavez presented a poster at the 2011 SACNAS meeting in San Jose.
- Nancy Scherich (UCLA, 2010).
Project: Grid digrams (VIGRE funded)
Paper: A simplification of grid equivalence, submitted to Involve.
Nancy Scherich has completed an MSc at Oregon State and is continuing her PhD at UCSB.
Glasgow Level 4 Projects
- Dylan Madden (2014)
Sian Griffiths (2013)
Neil Henderson (2013)
About this page
The page design for this website was stolen from my friend Patrick Ingram;
the photo was taken at Spanish Banks in Vancouver, early in the morning, by my friend Andrew Rowat.