The Poincare homology sphere and almost simple knots in lens spaces

TL;DR

This paper proves that certain L-space homology spheres arising from specific knots in lens spaces are always the Poincaré homology sphere, connecting knot theory, Heegaard Floer homology, and 3-manifold topology.

Contribution

It provides a simple proof that these L-space homology spheres are always the Poincaré homology sphere, clarifying their topological nature.

Findings

L-space homology spheres from these knots are always the Poincaré homology sphere.

The proof simplifies understanding of the relationship between knots in lens spaces and 3-manifold topology.

Connections between knot Floer homology and classical 3-manifold invariants are elucidated.

Abstract

Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S^3. Rasmussen shows that when the `left-handed' one is in the homology class of the dual to a Berge knot of type VII, it admits an L-space homology sphere surgery. In this note we give a simple proof that these L-space homology spheres are always the Poincar\'e homology sphere.