Some Computations on Instanton Knot Homology

TL;DR

This paper introduces an algorithm to compute upper bounds of instanton knot homology dimensions from knot diagrams, demonstrating sharp bounds for knots up to seven crossings and linking specific homology forms to instanton L-space knots.

Contribution

The authors develop a new algorithm for estimating instanton knot homology from diagrams and establish a connection between homology form and instanton L-space knots.

Findings

Algorithm provides sharp bounds for knots up to seven crossings.

The method effectively estimates instanton knot homology dimensions.

Certain homology forms imply the knot is an instanton L-space knot.

Abstract

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm that can compute an upper bound of the dimension of instanton knot homology from knot diagrams. We test the effectiveness of the algorithm and found that for all knots up to seven crossings, the algorithm provides sharp bounds. In the second half of the paper, we show that, if the instanton knot Floer homology of a knot has a specified form, then the knot must an instanton L-space knot.