Kauffman states and Heegaard diagrams for tangles

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 $\widehat{\mathrm{HFT}}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\widehat{\mathrm{HFT}}$ and prove symmetry relations for $\widehat{\mathrm{HFT}}$ of 4-ended tangles that echo those for ${\nabla }_T^s$.

version accepted for publication in Algebraic and Geometric Topology

References in corpus (5)

• The Alexander polynomial as quantum invariant of links
• On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology
• Combinatorial tangle Floer homology
• Peculiar modules for 4-ended tangles
• On Conway's potential function for colored links

Cited by in corpus (2)

• Thin links and Conway spheres
• L-space knots have no essential Conway spheres