Holomorphic discs and surgery exact triangles

TL;DR

This paper establishes a link between surgery exact sequences in knot Floer homology and holomorphic disk counts, enabling combinatorial computations of cobordism map ranks and refining existing sequences with orientations and spinc-structures.

Contribution

It introduces a new connection between surgery exact sequences and holomorphic disk moduli spaces, allowing for combinatorial rank calculations and sequence refinements.

Findings

The exact sequence in [18] applies with coherent orientations.

Ranks of cobordism maps can be computed combinatorially.

The work relates holomorphic disk counts to algebraic invariants.

Abstract

We show a connection between a surgery exact sequence in knot Floer homology and the sequence derived in [18]. As a consequence of this relationship we see that the exact sequence in [18] also works with coherent orientations and admits refinements with respect to spinc-structures. As an application of this discussion, we prove that the ranks of the image and kernel of certain cobordism maps between knot Floer homologies can be computed combinatorially by relating them to a count of certain moduli spaces of holomorphic disks.