Knot surgery formulae for instanton Floer homology II: applications

TL;DR

This paper advances the understanding of instanton Floer homology by developing new surgery formulas, applying them to various 3-manifolds, and characterizing almost L-space knots with specific properties.

Contribution

It introduces enhanced surgery formulas for instanton homology and applies them to classify and analyze properties of almost L-space knots.

Findings

Enhanced large surgery formula for null-homologous knots.

Classification of almost L-space knots with genus at least 2.

Identification of specific genus-one almost L-space knots as figure eight or mirror of 5_2.

Abstract

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous knots in any 3-manifold, and a formula encoding a large portion of $I^\sharp(S^3_0(K))$. Second, we use the integral surgery formula to study the framed instanton homology of many 3-manifolds: Seifert fibered spaces with nonzero orbifold degrees, especially nontrivial circle bundles over any orientable surface, surgeries on a family of alternating knots and all twisted Whitehead doubles, and splicings with twist knots. Finally, we use the previous techniques and computations to study almost L-space knots, ${\it i.e.}$, the knots $K\subset S^3$ with $\dim I^\sharp(S_n^3(K))=n+2$ for some $n\in\mathbb{N}_+$. We show that an almost L-space knot of genus at least $2$ is fibered and strongly quasi-positive, and a genus-one almost L-space knot must be either the figure eight or the mirror of the $5_2$ knot in Rolfsen's knot table.