Knots which admit a surgery with simple knot Floer homology groups

TL;DR

This paper investigates conditions under which surgeries on knots produce manifolds with simple Floer homology, establishing bounds related to the knot's genus and characterizing the resulting manifolds in specific cases.

Contribution

It provides a lower bound on the surgery coefficient for knots with simple Floer homology and characterizes the Floer homology of knots in the standard sphere when the resulting manifold is an L-space.

Findings

n >= 2g(K) for simple Floer homology after surgery

In S^3, the Floer homology of the knot is determined by its Alexander polynomial

Surgeries yielding L-spaces impose constraints on the knot's properties

Abstract

We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups of K are determined by its Alexander polynomial.