Kauffman states and Heegaard diagrams for tangles

TL;DR

This paper introduces polynomial invariants for tangles based on Kauffman states, explores their properties, and develops a categorified Heegaard Floer homology version, revealing symmetry relations and invariance under mutation.

Contribution

It defines new polynomial tangle invariants using Kauffman states and Alexander codes, and introduces a categorified Heegaard Floer homology for tangles, linking combinatorial and geometric approaches.

Findings

Proves symmetry relations for invariants of 4-ended tangles.

Shows multivariable Alexander polynomial invariance under Conway mutation.

Defines a bordered sutured invariant  for tangles with a bigrading.

Abstract

We define polynomial tangle invariants $\nabla_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for $\nabla_T^s$ of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants $\nabla_T^s$ can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of $\nabla_T^s$: a Heegaard Floer homology $\widehat{\operatorname{HFT}}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\widehat{\operatorname{HFT}}$ and prove symmetry relations for $\widehat{\operatorname{HFT}}$ of 4-ended tangles that echo those for $\nabla_T^s$.