The invariance of knot lattice homology

TL;DR

This paper proves that the filtered lattice chain homotopy type of a certain class of plumbed 3-manifolds with embedded knots remains invariant under diffeomorphisms, highlighting a key topological property.

Contribution

It establishes the invariance of the filtered lattice chain homotopy type for a specific class of plumbed 3-manifolds with knots, advancing understanding of their topological invariants.

Findings

Filtered lattice chain homotopy type is an invariant of the 3-manifold's diffeomorphism class.

The result applies to manifolds derived from negative-definite forests with one unframed vertex.

This invariance aids in classifying and distinguishing complex 3-manifolds with embedded knots.

Abstract

Assume $\Gamma$ is a negative-definite forest with exactly one unframed vertex, and $M(\Gamma)$ is the resulting plumbed 3-manifold with a knot embedded. We show that the filtered lattice chain homotopy type of $\Gamma$ is an invariant of the diffeomorphism type of $M(\Gamma)$.