A combinatorial spanning tree model for knot Floer homology

TL;DR

This paper introduces a new combinatorial spanning tree model for knot Floer homology, using spectral sequences derived from skein exact triangles, enabling algorithmic computation of knot invariants.

Contribution

It develops the first combinatorial spanning tree framework for knot Floer homology, connecting spectral sequences with spanning trees for computational purposes.

Findings

Spectral sequence converges to a stabilized delta-graded knot Floer homology.

The (E_2, d_2) page is an algorithmically computable chain complex.

No higher differentials beyond d_2 are present.

Abstract

We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over $\mathbb{Z}/2\mathbb{Z}$. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The $(E_2,d_2)$ page of this spectral sequence is an algorithmically computable chain complex expressed in terms of spanning trees, and we show that there are no higher differentials. This gives the first combinatorial spanning tree model for knot Floer homology.