Knot Floer homology and integer surgeries

TL;DR

This paper describes how to compute the Heegaard Floer homology of integer surgeries on knots in three-manifolds using knot invariants, and applies this to circle bundles over Riemann surfaces.

Contribution

It provides a new method to determine Heegaard Floer homology of surgeries based on knot invariants, extending previous techniques.

Findings

Computed Heegaard Floer homology for non-trivial circle bundles over Riemann surfaces.

Established a relationship between knot invariants and surgery homology.

Demonstrated the method with explicit calculations.

Abstract

Let $K$ be a null-homologous knot in a three-manifold $Y$. We give a description of the Heegaard Floer homology of integer surgeries on $Y$ along $K$ in terms of the filtered homotopy type of the knot invariant for $K$. As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles over Riemann surfaces.