“Transverse knots distinguished by knot Floer homology”, with L. Ng and P. Ozsváth, Journal of Symplectic Geometry, in press, 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.
“3-manifolds efficiently bound 4-manifolds”, with F. Costantino, Journal of Topology, 1 (2008), 703–745, 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.
“Legendrian knots, transverse knots and combinatorial Floer homology”, with P. Ozsváth and Z. Szabó, Geometry and Topology, 12 (2008), 941–980, 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.
“On combinatorial link Floer homology”, with C. Manolescu, P. Ozsváth, and Z. Szabó, Geometry and Topology, 11 (2007), 2339–2412, 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.
“Cluster algebras and triangulated surfaces. Part I: Cluster complexes”, with S. Fomin and M. Shapiro, Acta Mathematica, to appear, 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.
“A random tunnel number one 3-manifold does not fiber over the circle”, with N. Dunfield, Geometry and Topology, 10 (2006), 2431–2499, 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.
“One-forms on meshes and applications to 3D mesh parametrization”, with S. Gortler and C. Gotsman, Computer Aided Geometric Design, 23, 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.
“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), 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, 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.
“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.
“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".
“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, 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, 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, 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.
“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. 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.
“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.
“Bordered Heegaard Floer homology: Invariance and pairing”, with R. Lipshitz and P. Ozsváth, arXiv:0810.0687. An extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. We associate a DGA to every connected surface F, and a module over F to a manifold with boundary F, in a way that lets us reconstruct the invariant of the closed manifold.
“Slicing planar grid diagrams: A gentle introduction to bordered Heegaard Floer homology”, with R. Lipshitz and P. Ozsváth, 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 as described above. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications.
“Characterizing generic global rigidity”, with S. Gortler and A. Healy, 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.
“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.
“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.
“Cluster algebras and triangulated surfaces. Part II: Lambda lengths”, with S. Fomin. 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.
"The F4 and E6 families have only a finite number of points" (PDF, Source). Computations giving evidence against Deligne's conjecture on the existence of an exceptional series of Lie algebras. (Version of 2004-11-13)
"A shadow calculus for 3-manifolds (PDF), 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" (PDF). 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.)
"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.