On lens space surgeries from the Poincar\'e homology sphere

TL;DR

This paper develops a lattice embedding obstruction to identify which lens spaces can be obtained via surgery on knots in the Poincaré homology sphere, revealing restrictions on such surgeries and classifying specific cases.

Contribution

It introduces a new lattice embedding obstruction based on Greene's changemaker lattices, advancing the understanding of lens space surgeries in the Poincaré homology sphere.

Findings

If a knot in the Poincaré homology sphere yields a lens space with p ≥ 2g(K), then an equivalent surgery exists on a Tange knot.

The only knots in the Poincaré homology sphere admitting half-integer lens space surgeries are two specific cases.

Lens space surgeries on hyperbolic knots in the Poincaré homology sphere are proven to be integral.

Abstract

Building on Greene's changemaker lattices, we develop a lattice embedding obstruction to realizing an L-space bounding a definite 4-manifold as integer surgery on a knot in the Poincar\'e homology sphere. As the motivating application, we determine which lens spaces are realized by $p/q$-surgery on a knot $K$ when $p/q > 2g(K) -1$. Specifically, we use the lattice embedding obstruction to show that if $K(p)$ is a lens space and $p \geq 2g(K)$, then there exists an equivalent surgery on a Tange knot with the same knot Floer homology groups; additionally, using input from Baker, Hedden, and Ni, we identify the only two knots in the Poincar\'e homology sphere that admit half-integer lens space surgeries. Thus, together with the Finite/Cyclic Surgery Theorem of Boyer and Zhang, we obtain the corollary that lens space surgeries on hyperbolic knots in the Poincar\'e homology sphere are integral.