We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n, n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n, n) Springer variety. We also give a formula for comultiplication in the Bar-Natan… (More)
The sl 3 spider is a diagrammatic category used to study the representation theory of the quantum group Uq(sl 3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov-Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm.… (More)
Consider the following question. Given a suitable linear series L on a smooth surface S, how many curves in L have a given analytic or topological type of singularity? By " suitable " linear series with respect to a type of singularity, we mean that there are finitely many curves with the singularity in the linear series and their codimension is maximal.… (More)
Springer varieties are studied because their cohomology carries a natural action of the symmetric group S n and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties X n as subvarieties of the product of spheres (S 2) n. We show that if X n is embedded antipodally in (S 2) n… (More)
We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are 'odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full… (More)
Every link in R 3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.
Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety. In this thesis I give a geometric proof that the cohomology of the flag variety surjects onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the… (More)
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n, n, n]: the reduced web basis associated to Kuperberg's combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n, n], the spider category is the… (More)