Publications
I put all my papers and preprints on the arXiv. This links to my arXiv papers. Below I give a very brief description of my publications.
Preprints:
An operad for splicing describes a new topological operad that encodes splicing of knots in the 3-dimensional case. The space of long knots in R^3 is shown to be "free" over this operad with free generating subspace the torus and hyperbolic knots. The splicing operad also has a relatively simple homotopy-type in this case.
Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census. This paper explores the question of which 3-manifolds smoothly embed in the 4-sphere, where the terrain of exploration is the census of 3-manifolds that admit semi-simplicial triangulations with 11 or less tetrahedra.
Refereed publications:
Topology of spaces of knots in dimension 3, to appear in Proc. Lond. Math. Soc. This paper describes the homotopy-type of the space of smooth embeddings of a circle in the 3-sphere. The homotopy-type of each path-component is given by an iterated bundle construction which is determined by the JSJ-decomposition of the knot complement.
An obstruction to a knot being deform-spun via Alexander polynomials (with Alexandra Mozgova) Proc. Amer. Math. Soc. 137 (2009), 3547-3552. This paper points out that Alexander polynomials give obstructions to knots being deform-spun.
On the homology of the space of knots. (with Fred Cohen) Geometry and Topology. Vol 13 (2009) 99--139. The rational homology of the space of long knots in R^3 is shown to be a free Poisson algebra. We also find torsion of all orders in the integral homology of the space of long knots in R^3, and give a homological characterization of the unknot component in both the space of long knots and the space of embeddings of S^1 in S^3.
A family of embedding spaces. Geometry and Topology Monographs 13 (2008), 41-83. This paper studies the space of embeddings of one sphere in another. There is a related long embedding space of Euclidean spaces, and this paper studies what is known about the iterated loop-space structures on those spaces.
The operad of framed discs is cyclic. Journal of Pure and Applied Algebra 212 no. 1, (2008) 193--196. This is a short argument that the operad of framed little n-discs is a cyclic operad.
Little cubes and long knots. Topology. 46 (2007) 1--27. Little cube operads are shown to act on various spaces of long knots. The space of long knots in R^3 is shown to be a free little 2-cubes object over the subspace of prime knots.
JSJ-decompositions of knot and link complements in the 3-sphere. L'enseignement Mathe'matique (2) 52 (2006), 319--359. This paper gives a bijective correspondence between the isotopy classes of oriented knots and links in S^3 and a class of labeled, acyclic trees. Roughly, this is a `uniqueness theorem' for Schubert's satellite decomposition of knots. Closely related is Bonahon and Siebenmann's almost-published paper New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots. Bonahon and Siebenmann have an abbreviated discussion of the splice decomposition. The main point of their paper (from my point of view) is that there is a further and very pleasant description of the JSJ-decomposition of the 2-sheeted branched cover of the 3-sphere branched over the knot which is frequently very useful for computing things like the symmetry groups of knots, allowing one to bypass SnapPea.
New Perspectives on Self-Linking. (with Jim Conant, Kevin Scannell and Dev Sinha) Advances in Mathematics. 191 (2005) 78--113. A direct relation between the geometry of a knot and the z^2 coefficient of the Alexander-Conway polynomial of the knot is constructed. We hint at further possible connections.
Quadrisecant pictures and animations related to the above self-linking paper.
On the image of the Lawrence-Krammer representation. J. Knot. Thry. Ram. Vol 14. No. 6. (2005) 1-17. The Lawrence-Krammer representation is shown to be unitary, and it is shown that the conjugacy problem in the image of the Lawrence-Krammer representation is quite different from the conjugacy problem in braid groups.
Here is a sketch (intended for the paper) of the signature of the Hermitian form vs the (q,t) variables, for the 6-stranded braid group.
The mapping class group of a genus 2 surface is linear. (with Stephen Bigelow) Algebr. Geom. Topol. 1 (2001), no. 34. 699--708. We construct a rank 64 faithful representation of the mapping class group of a genus 2 surface.
Other:
My CV: cv.pdf