“**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(d ^{2})*,
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 S^{3}**”, with
D. Bar-Natan, S. Garoufalidis, and L. Rozansky,

“

“

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.

“**Cluster algebras and
triangulated surfaces. Part II: Lambda lengths**”, with
S. Fomin, arXiv:1210.5569.
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.

“**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.

"**From rubber bands to
rational maps: Research report**". Draft of April 19, 2014.
Among other things, a new characterization of which branched
self-covers of the sphere are equivalent to rational maps.

"**The F_{4} and E_{6} 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.

Dylan Thurston