Research

I study quantum and low-dimensional topology and related topics in representation theory and mathematical physics. In particular, I am interested in connections between quantum invariants of links and 3-manifolds and hyperbolic geometry.

Holonomy invariants

An invariant takes as data a topological object $X$ (like a knot or knot complement) and produces a simpler object, like a number or polynomial. A holonomy invariant also depends on a map from $\pi_1(X)$ to a Lie group, such as $\mathrm{SL}_2(\mathbb C)$, which we can think of as the holonomy of a flat $\mathfrak{sl}_2$-connection.

Classical invariants of this type include Reidemeister torsion/Alexander polynomials (both ordinary and twisted) and geometric invariants like hyperbolic volume. Several authors have defined new quantum holonomy invariants that extend known quantum constructions (like the colored Jones polynomials) to quantum holonomy invariants.

I showed that one construction of this type (a sort of deformed Jones polynomial) gives the twisted Reidemeister torsion of the link complement. Right now I’m working on extensions of this result.

Graphs in surfaces

Less recently, Kyle Miller and I wrote a paper on extending some quantum-topology techniques for planar graphs to graphs embedded in higher-genus surfaces. In particular, we found a state-sum invariant of embedded graphs called the “S-polynomial.” It should be counting something (as the flow and chromatic polynomials do) but we’re not sure what. It may have to do with “defects” in lattice models.