Combinatorial tangle Floer homology

TL;DR

This paper develops a combinatorial approach to tangle Floer homology, extending bordered Floer homology to knots and links in various 3-manifolds, providing new invariants and connections to knot Floer homology.

Contribution

It introduces a combinatorial construction of tangle invariants in multiple 3-manifolds, generalizing bordered Floer homology to a broader class of objects.

Findings

Constructed gluable combinatorial invariants for tangles in S^3, D^3, and I×S^2.

Reproduces a stabilized version of knot Floer homology for S^3.

Provides a new framework for studying knots and links via combinatorial methods.

Abstract

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.