A quantum categorification of the Alexander polynomial

in Geometry and Topology (2021)

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page ... [more ▼]

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky. [less ▲]

Frobenius Extension II

E-print/Working paper (2020)

The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories ... [more ▼]

The first two sections of the paper provide a convenient scheme and additional diagrammatics for working with Frobenius extensions responsible for key flavors of equivariant SL(2) link homology theories. The goal is to clarify some basic structures in the theory and propose a setup to work over sufficiently non-degenerate base rings. The third section works out two related SL(2) evaluations for seamed surfaces. [less ▲]

Box-Total Dual Integrality, Box-Integrality, and Equimodular Matrices

in Mathematical Programming (2020)

A polyhedron is box-integer if its intersection with any integer box {ℓ≤x≤u} is integer. We define principally box-integer polyhedra to be the polyhedra P such that kP is box-integer whenever kP is ... [more ▼]

A polyhedron is box-integer if its intersection with any integer box {ℓ≤x≤u} is integer. We define principally box-integer polyhedra to be the polyhedra P such that kP is box-integer whenever kP is integer. We characterize them in several ways, involving equimodular matrices and box-total dual integral (box-TDI) systems. A rational r×n matrix is equimodular if it has full row rank and its nonzero r×r determinants all have the same absolute value. A face-defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron. Box-TDI systems are systems which yield strong min-max relations, and the underlying polyhedron is called a box-TDI polyhedron. Our main result is that the following statements are equivalent. - The polyhedron P is principally box-integer. - The polyhedron P is box-TDI. - Every face-defining matrix of P is equimodular. - Every face of P has an equimodular face-defining matrix. - Every face of P has a totally unimodular face-defining matrix. - For every face F of P, lin(F) has a totally unimodular basis. Along our proof, we show that a cone {x:Ax≤0} is box-TDI if and only if it is box-integer, and that these properties are passed on to its polar. We illustrate the use of these characterizations by reviewing well known results about box-TDI polyhedra. We also provide several applications. The first one is a new perspective on the equivalence between two results about binary clutters. Secondly, we refute a conjecture of Ding, Zang, and Zhao about box-perfect graphs. Thirdly, we discuss connections with an abstract class of polyhedra having the Integer Carathéodory Property. Finally, we characterize the box-TDIness of the cone of conservative functions of a graph and provide a corresponding box-TDI system. [less ▲]