Kauffman states, bordered algebras, and a bigraded knot invariant

TL;DR

This paper introduces a new bigraded knot invariant derived from Kauffman states and bordered algebras, which relates closely to knot Floer homology and generalizes classical invariants like the Alexander polynomial.

Contribution

It constructs a novel homological invariant using algebraic structures from Kauffman states and diagram decompositions, linking diagrammatic and algebraic approaches in knot theory.

Findings

The invariant's Euler characteristic equals the Alexander polynomial.

The construction connects Kauffman states with bordered algebra modules.

Provides a new algebraic framework for knot invariants.

Abstract

We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.