I put all my papers and preprints on the arXiv. Below I give a brief description of them.


Bi-filtrations and persistence paths for 2-Morse functions. (with Tomasz Kaczynski)

This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description of the multi-graded persistent homology associated to such filtrations. The main result of the paper is a description of the evolution of the bi-filtration of f in terms of cellular attachments. An analogy of Morse-Conley equation and Morse inequalities along so called persistence paths are derived. A scheme for computing path-wise barcodes is proposed.

Knotted 3-balls in S^4. (with David Gabai)

In this paper we describe the mapping class group of "the barbell manifold" and what it tells us about smooth isotopy of 3-manifolds in some small 4-manifolds.  Specifically, the "barbell" is the (4,2)-handlebody of genus 2, i.e. the boundary connect-sum of two copies of S^2 x D^2. We show that the mapping class group of the barbell manifold, i.e. \pi_0 Diff(Barbell), where the diffeomorphisms fix the boundary pointwise, is infinite cyclic (after modding out by the mapping class group of D^4). We then consider embedding the barbell into various 4-manifolds, and the question of whether or not the natural extension of the barbell diffeomorphism is isotopically trivial in these 4-manifolds.  From this we can conclude that the mapping class groups of both S^1 x D^3 and S^1 x S^3 are not finitely generated. For S^1 x D^3 the idea of the proof is to show these diffeomorphisms act non-trivially on the isotopy classes of reducing 3-balls, i.e. show f({1}xD^3) is not isotopic to {1}xD^3.  To do this, we imagine D^3 as a 2-parameter family of intervals, thus f({1}xD^3) can be viewed as producing an element of the 2nd homotopy group of the space of smooth embeddings of an interval in S^1 x D^3. The core of the proof involves developing an invariant that can detects the homotopy groups of such embedding spaces. These can be thought of as Vassiliev invariants.

Refereed publications

Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census. (with Ben Burton)

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. Experimental Mathematics, Apr. 2020.

Combinatorial spin structures on triangulated manifolds.

This paper gives a description of spin and spinc structures on triangulated manifolds in a combinatorial language suitable for computer implementation. The primary novelty in the approach is the use of naturality of binary symmetric group constructions to avoid elaborate constructions with explicit smoothings of PL manifolds. DOI: 10.2140/agt.2018.18.1259

Embedding calculus knot invariants are of finite type.

This gives a reason the knot-invariants coming from Embedding Calculus are finite-type invariants. We show that path-components of the Taylor tower have an abelian group structure, compatible with concatenation of knots, and with respect to this group structure these invariants are finite-type. With Robin Koytcheff, Dev Sinha and Jim Conant. Algebraic and Geometry Topology, September 2016.

A small, infinitely-ended 2-knot group.

Jon Hillman and I show that a 2-knot group discovered in the course of a census of 4-manifolds with small triangulations (which itself is ongoing work with Ben Burton) is an HNN extension with finite base and proper associated subgroups, and has the smallest base among such knot groups. J. Knot Thry. Ram. Vol. 26, Issue 1.

Topology of musical data.

Bill Sethares and I apply and interpret persistent homology to data directly taken from music. J. Math & Music. Volume 8, Issue 1, January 2014, pages 73-92.

Triangulating a Cappell-Shaneson knot complement.

Mathematical Research Letters 19 (2012), no. 5, 1117-1126. We show that one of the Cappell-Shaneson knot complements admits an extraordinarily small triangulation, containing only two 4-dimensional simplices. With Ben Burton and Jon Hillman. Note: Ahmad Issa (2017) found a natural way of finding PL-compatible triangulations of Cappell-Shaneson exteriors in his M.Phil thesis, and our triangulation with two 4-dimensional simplices is generated via his technique. Issa's work could be seen as the natural generalization of the Floyd-Hatcher triangulations of punctured torus bundles over the circle.

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 R3 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, being the free product of a 2-cubes operad together with a free operad on a space (with a certain group action) which via splicing encodes cabling and hyperbolic splicing operations. Journal of Topology 2012; doi: 10.1112/jtopol/jts024

Topology of spaces of knots in dimension 3.

Proc. Lond. Math. Soc. Vol 101 (2) Sept 2010. 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 R3 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 R3, and give a homological characterization of the unknot component in both the space of long knots and the space of embeddings of S1 in S3.

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 R3 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 S3 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 z2 coefficient of the Alexander-Conway polynomial of the knot is constructed. We hint at further possible connections.

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.