Papers and Preprints

The papers and preprints are organized by topic.

Bordered Floer homology

Bordered Heegaard Floer homology: Invariance and pairing”, with R. Lipshitz and P. Ozsváth, Memoirs of the AMS, accepted for publication, arXiv:0810.0687. An extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. We associate a differential graded algebra A to every connected surface F, and a module over A to a manifold with boundary F, in a way that lets us reconstruct the invariant of the closed manifold.

Bimodules in bordered Heegaard Floer homology”, with R. Lipshitz and P. Ozsváth, Geometry and Topology, 19 (2015), 525–724, arXiv:1003.0598. We extend bordered Floer homology to allow for 3-manifolds with two boundary components, algebraically giving bimodules. In particular, this allows for reparametrization of the surface.

A tour of bordered Floer theory”, with R. Lipshitz and P. Ozsváth, Proceedings of the National Academy of Sciences, 108(20):8085--8092, May 17, 2011 arXiv:1107.5621. A summary of the major structures and results of bordered Floer theory.

Slicing planar grid diagrams: A gentle introduction to bordered Heegaard Floer homology”, with R. Lipshitz and P. Ozsváth, In Proceedings of the Gökova Geometry-Topology Conference 2008, arXiv:0810.0695. A description of some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for bordered Heegaard Floer homology. This paper is meant to serve as an introduction to the subject, and does not itself have immediate topological applications.

Heegaard Floer homology as morphism spaces”, with R. Lipshitz and P. Ozsváth, Quanum Topology, 2(4):381--449, 2011 arXiv:1005.1248. Another version of the pairing theorem for bordered Floer homology, using homomorphisms rather than tensor products. This allows a more direct comparison with Fukaya-category constructions, and leads to many new dualities.

A faithful linear-categorical action of the mapping class group of a surface with boundary”, with R. Lipshitz and P. Ozsváth, Journal of the European Mathematical Society, 15(4):1279--1307, 2013, arXiv:1012.1032. We show that the categorical action of the based mapping class group given by bordered Floer homology is faithful. This paper is partly intended as an introduction for geometric topologists interested in the theory.

Computing HF^ by factoring mapping classes”, with R. Lipshitz and P. Ozsváth, Geometry and Topology, 18 (2014), 2547–2681, arXiv:1005.2550. We explicitly compute the bordered Floer homology bimodules associated to arc-slides, and use them to give a combinatorial description of HF^ of a closed 3-manifold, as well as the bordered Floer homology of any 3-manifold with boundary.

Bordered Floer homology and the spectral sequence of a branched double cover I”, with R. Lipshitz and P. Ozsváth, Journal of Topology, 7 (2014), 1155–1199, arXiv:1011.0499. We explicitly calculate the spectral sequence starting at Khovanov homology of a link and converging to the Heegaard Floer homology of its branched double cover. In this paper we compute an explicit combinatorial spectral sequence; in the sequel we show that this spectral sequence agrees with the previously known one.

Notes on bordered Floer homology”, with R. Lipshitz and P. Ozsváth, In Contact and Symplectic Topology, 275–355, Bolyai Mathematical Studies 26, 2014, arXiv:1211.6791. A series of notes for lectures given by Robert Lipshitz at the CaST conference in Budapest in the summer of 2012.

Relative ℚ-gradings from bordered Floer theory”, with R. Lipshitz and P. Ozsváth. Turkish Journal of Mathematic, accepted for publication, arXiv:1211.6990. We show how to reconstruct the relative ℚ-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.

Bordered Floer homology and the spectral sequence of a branched double cover II: The spectral sequences agree”, with R. Lipshitz and P. Ozsváth, Journal of Topology, 9 (2016), 607–686, arXiv:1404.2894. We show that the spectral sequence we constructed earlier, from the Khovanov homology of a knot to its Heegaard Floer homology, agrees with the previously-defined one using holomorphic curves.

Other papers on Heegard Floer homology

On combinatorial link Floer homology”, with C. Manolescu, P. Ozsváth, and Z. Szabó, Geometry and Topology, 11 (2007), 2339–2412, arXiv:math.GT/0610559. We give an elementary explanation of the basic properties of the combinatorial Floer homology introduced by Manolescu-Ozsváth-Sarkar, including a self-contained proof of its invariance. We also extend the theory to work with signs, over ℤ rather than ℤ/2ℤ.

Legendrian knots, transverse knots and combinatorial Floer homology”, with P. Ozsváth and Z. Szabó, Geometry and Topology, 12 (2008), 941–980, arXiv:math.GT/0611841. We use the explicit chain maps used in the proof of invariance of link Floer homology to construct invariants of Legendrian and transverse knots, with values in the combinatorial Floer homology of the underlying topological knot.

Transverse knots distinguished by knot Floer homology”, with L. Ng and P. Ozsváth, Journal of Symplectic Geometry, 6(2008):461–490, arXiv:math.GT/0703446. We use the invariant of transverse knots above to find several new examples of distinct transverse knots with the same classical invariants.

Grid diagrams and Heegaard Floer invariants”, with C. Manolescu and P. Ozsváth, arXiv:0910.0078. We use a link surgery spectral sequence due to Manolescu and Ozsváth to give a combinatorial description of Heegaard Floer homology for arbitrary three-manifolds.

Naturality and mapping class groups in Heegaard Floer homology”, with A. Juhász, arXiv:1210.4996. We show that all flavors of Heegaard Floer homology are natural, in the sense that they assign a group to a pointed 3-manifold, not just an isomorphism class of groups.

Conformal geometry and rational maps

From rubber bands to rational maps: Research report”, Research in the Mathematical Sciences, 3:15, 2016, doi:10.1186/s40687-015-0039-4, arXiv:1502.02561. A new positive characterization of which branched self-covers of the sphere are equivalent to rational maps. See also the program Harmonic below.

Conformal surface embeddings and extremal length”, with J. Kahn and K. Pilgrim. Preprint, 2015, arXiv:1507.05294. A characterization of when one Riemann surface conformally embeds inside another in a given homotopy class of maps, using extremal length. This is used in the characterization of which branched self-covers are equivalent to rational maps.

Elastic Graphs”. Preprint, 2016, arXiv:1607.00340. When is one elastic graph looser than another, regardless of which target space you consider? This paper answers one version of this question, and also introduces a family of energies for maps between graphs. This is used in the characterization of which branched self-covers are equivalent to rational maps.

A positive characterization of rational maps”. Preprint, 2016, arXiv:1612.04424. When is a topological branched self-cover of the sphere equivalent to a rational map on CP1? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic spine that gets "looser" under backwards iteration.

Geometric topology

The algebra of knotted trivalent graphs and Turaev's shadow world”, in Geometry and Topology Monographs, Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001), T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts and V Turaev (editors), arXiv:math.GT/0311458. The algebra of knotted trivalent graphs may be thought of as a generalisation of many different ways of representing knots. This paper introduces the algebra of knotted trivalent graphs and explains the connection to Turaev's shadow world diagrams.

With Ian Agol, Appendix to “The volume of hyperbolic alternating link complements” by Marc Lackenby, Proc. London Math. Soc. (3), 88 (2004), no. 1, 204–224, arXiv:math.GT/0012185. In his paper, Marc Lackenby proves that the volume of the complement of a hyperbolic alternating link is bounded above and below by linear functions of the twist number, the number of non-parallel crossings. In the appendix, Ian Agol and I improve the upper bound and show that it is asymptotically sharp by constructing an explicit chain-link fence link.

A random tunnel number one 3-manifold does not fiber over the circle”, with N. Dunfield, Geometry and Topology, 10 (2006), 2431–2499, arXiv:math.GT/0510129. A proof that, in a measured lamination model, a random 3-manifold does not fiber over the circle. One motivation is to give insight into the Virtual Fibration Conjecture. The source code is also available.

3-manifolds efficiently bound 4-manifolds”, with F. Costantino, Journal of Topology, 1 (2008), 703–745, arXiv:math.GT/0506577. A proof that an oriented 3-manifold of complexity d bounds a 4-manifold of complexity O(d2), where complexity measured suitably. In particular, this implies that surgery diagrams are not too inefficient as a way of representing 3-manifolds. The proof uses the technology of shadow surfaces.

Grid diagrams, braids, and contact geometry”, with L. Ng, in Proceedings of the Gökova Geometry-Topology Conference 2008, arXiv:0812.3665. We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.

Non-peripheral ideal decompositions of alternating knots”, with S. Garoufalidis and I. Moffatt, arXiv:1610.09901. We use the small cancellation property of the Dehn presentation of alternating knot groups to show that several reasonable ideal decompositions of alternating knot complements are not peripheral.

A shadow calculus for 3-manifolds” (draft), with F. Costantino. Two versions of a finite set of moves that relate all shadow surfaces representing the same 3-manifold (with slightly different assumptions). The corresponding question for 4-manifolds is related to the Andrews-Curtis conjecture.

"On geometric intersection of curves in surfaces" (draft). An exploration of how curves on surfaces intersect. Among other results, I derive simple formulas for the transformation of Dehn-Thurston coordinates when you change the pair of pants decomposition. (R. Penner had earlier found more complicated formulas for the same problem.)

Finite type and quantum invariants

Integral expressions for the Vassiliev knot invariants”, senior thesis, Harvard University, May 1995, math.QA/9901110. Advisor: R. Bott. Building on earlier work of Bott and Taubes, I construct here a universal Vassiliev invariant using the topology of configuration spaces.  This is an alternative to the earlier Kontsevich integral, which is less topological.

Wheels, wheeling, and the Kontsevich integral of the unknot”, with D. Bar-Natan, S. Garoufalidis, and L. Rozansky, Israel J. Math. 119 (2000), 217–237, q-alg/9703025. Here we make the “Wheels” and “Wheeling” conjectures for finite type invariants. They are motivated by some facts on the level of Lie algebras; we prove the conjectures on this level. The conjectures involve the Bernoulli numbers in an intimate way.

The Aarhus invariant of rational homology 3-spheres I: A highly non-trivial flat connection on S3”, with D. Bar-Natan, S. Garoufalidis, and L. Rozansky, Selecta Mathematica (N.S.) 8 (2002), no. 3, 315–339, arXiv:q-alg/9706004.
The Aarhus invariant of rational homology 3-spheres II: Invariance and universality”, with D. Bar-Natan, S. Garoufalidis, and L. Rozansky, Selecta Mathematica (N.S.) 8 (2002), no. 3, 341–371,  arXiv:math.QA/9801049.
The Aarhus invariant of rational homology 3-spheres III: The relation with the Le-Murakami-Ohtsuki invariant”, with D. Bar-Natan, S. Garoufalidis, and L. Rozansky, Selecta Mathematica (N.S.) 10 (2004), no. 3, 305–325, arXiv:math.QA/9808013.
These three papers introduce the “Aarhus integral”, a method for turning a universal finite-type invariant for knots into a universal finite-type invariant for 3-manifolds, using a diagrammatic version of integration. The first paper is introduction and intuition. The second we give precise definitions and prove that we construct a universal finite type invariant. In the third paper we show that this construction is equivalent to an earlier construction by Le, Murakami, and Ohtsuki; our contribution is motivation, and, in particular, a way of performing calculus on diagrams.

On the existence of finite type link homotopy invariants”, with Blake Mellor, J. Knot Theory Ramifications 10 (2001), no. 7, 1025–1039, math.GT/0010206.  We show that there are finite type link homotopy invariants for links with 9 or more components, but none for links with 5 or fewer components.

Wheeling: A diagrammatic analogue of the Duflo isomorphism”, Ph.D. thesis, U.C. Berkeley, math.QA/0006083.  Parts are joint work with Dror Bar-Natan and Thang Le.  Here we prove our earlier “Wheels” and “Wheeling” conjectures. The “Wheels” conjecture computes the Kontsevich integral of the unknot, the first knot for which this invariant is known to all degrees, and the “Wheeling” conjecture is a stronger, diagrammatic, analogue of the Duflo isomorphism.  Our proof boils down to properly interpreting some  trivial topological facts, analogous to “1+1=2” and “n*0=0”.

Two applications of elementary knot theory to Lie algebras and Vassiliev invariants”, with D. Bar-Natan and T. Le, Geometry and Topology, 7 (2003), no. 1, 1–31, math.QA/0204311. This is the published version of the parts of my Ph.D. thesis with the proof of the Wheels and Wheeling conjectures.

Perturbative 3-manifold invariants by cut-and-paste topology”, with G. Kuperberg, UC Davis Math 1999-36, math.GT/9912167. In this paper we construct a universal finite-type invariant for rational homology spheres using configuration spaces.  The construction is not particularly new; the new part is the proof that the invariant is universal (finite type with the correct weight system). In particular, this construction gives a new, elementary definition of the Casson invariant.

The F4 and E6 families have only a finite number of points” (draft). Computations giving evidence against Deligne's conjecture on the existence of an exceptional series of Lie algebras. (Version of 2004-11-13)

Cluster algebras and representation theory

Cluster algebras and triangulated surfaces. Part I: Cluster complexes”, with S. Fomin and M. Shapiro, Acta Mathematica, 201 (2008), 83–146, arXiv:math.RA/0608367. We study the cluster algebras arising from Teichmüller theory of bordered surfaces, first introduced by Gekhtman-Shapiro-Vainshtein and Fock-Goncharov. We describe these cluster algebras explicitly in terms of tagged triangulations and show that they give a large family of cluster algebras which are “mutationally finite”: although they are not finite type, there are only a finite number of combinatorial types of coefficients. We furthermore determine their homotopy type and growth rate.

Cluster algebras and triangulated surfaces. Part II: Lambda lengths”, with S. Fomin, Memoirs of the AMS, accepted for publication, arXiv:1210.5569. We explain the geometry behind the construction of cluster algebras from tagged triangulations.

Positive basis for surface skein algebras”, Proceedings of the National Academy of Sciences, 111 (2014), 9725–9732, arXiv:1310.1959. We give a natural basis for a certain skein algebra of a surface, corresponding to the twisted SL2 representations of the fundamental group. This basis is strongly positive, in the sense that the structure constants for multiplication are positive integers.

The complex volume of SL(n,ℂ) representations of 3-manifolds”, with S. Garoufaldis and C. Zickert, Duke Mathematical Journal, 164 (2015), 2099–2160, arXiv:1111.2828. We use Ptolemy coordinates (related to cluster algebras) to parametrize representations of a 3-manifold into SL(n,ℂ) and compute their volumes.

From dominoes to hexagons”, math.CO/0405482. A generalisation of domino tilings to topological tilings by hexagons, with connections to planar algebras, Legendrian knots, and cluster algebras.

Discrete geometry

Discrete one-forms on meshes and applications to 3D mesh parametrization”, with S. Gortler and C. Gotsman, Computer Aided Geometric Design, 23 (2006), 83–112, Harvard Computer Science TR-12-04. We use a discrete version of 1-forms on surfaces to give an easy proof of Tutte's theorem and generalize it to other contexts, including mesh decompositions of the torus.

Characterizing generic global rigidity”, with S. Gortler and A. Healy, American Journal of Mathematics, 132 (2010), no. 4, 897–939, arXiv:0710.0926. We prove of a conjecture by R. Connelly on which graphs are generically globally rigid. That is, for which graphs does a random rigid straight-line embedding in d dimensions have an alternate embedding with the same edge lengths? They can be characterized by a local and efficiently checkable criterion.

Sensor network localization using sensor perturbation”, with Y. Zhu and S. Gortler, Transactions on Sensor Networks, 7 (2011), no. 4, Article 36.

We apply some of the techniques of global rigidity to give a concrete method for “localizing” a sensor network, finding the positions of the nodes given only distance data.

On affine rigidity”, with S. Gortler, C. Gotsman, and L. Liu, Journal of Computational Geometry, 4 (2013), no. 1, 160–181, arXiv:1011.5553.

We study when networks can be determined by giving affine relations between the vertices, as opposed to the more usual Euclidean distances.

Generic global rigidity in complex and pseudo-Euclidean spaces”, with S. Gortler, in Rigidity and Symetry, Fields Institute Comunications 70, 131–154, 2014, arXiv:1212.6685.

We extend many of the results of the theory of global rigidity to situations when the network is complex or in a non-definite metric (as in hyperbolic space).

Characterizing the universal rigidity of generic frameworks”, with S. Gortler, Discrete and Computational Geometry, 51 (2014), no. 4, 1017–1036, arXiv:1001.0172.

We characterize when a generic framework is universally rigid; that is, it is rigid, and remains rigid when re-embedded in a space of any higher dimension.

Measurement isomorphism of graphs”, with S. Gortler, arXiv:1212.6551.

We define a notion of “d-measurement isomorphism” of graphs, characterizing when the set of possible squared edge lengths of two graphs are the same. In this note, we show that this property coincides with the 2-isomorphism property studied by Whitney.

Other papers

A bulk inflaton from large volume extra dimensions”, with B. Greene, D. Kabat, and J. Levin, Physics Letters B, 694 (2011), no. 4, 485–490, arXiv:1212.6551.

"Markup optimisation by probabilistic parsing", with Chung-chieh Shan. First-place winner in the ACM International Conference on Functional Programming programming contest. We solved the problem of markup optimisation in a stripped-down version of HTML by reinterpreting it as a parsing problem and applying standard techniques for optimal parsing.


Harmonic, a Haskell program for computing harmonic maps of graphs and stretch factors. This gives a criterion for when a given branched self-cover of the sphere is equivalent to a rational map. This program is in a very preliminary state, and currently has no user interface.


This section has short notes on various topics. Mostly they do not have enough content to be published as-is, although they may appear as part of larger works.

Discrete quadratic differentials, a model for a discrete version of quadratic differentials and measured foliations on a surface. Unlike other representations (e.g., with train tracks), this gives a uniform system of coordinates for all measured foliations on a closed surface simultaneously. It is also useful for approximating actual harmonic measured foliations.

Extremal Length on Curves and Graphs, concrete estimates relating a notion of “extremal length” on ribbon graphs to standard extremal length on a thickening of the graph.

Dylan Thurston