Invariance and naturality of knot lattice homology and homotopy

TL;DR

This paper proves that knot lattice homology is an invariant of the smooth knot type in rational homology spheres and explores its relation to topological and homotopy invariants of algebraic links.

Contribution

It establishes the invariance of knot lattice homology under smooth knot isotopy and relates it to the topology of minimal plumbing resolutions for algebraic links.

Findings

Knot lattice homology is an invariant of smooth knot types.

Knot lattice homology can be realized as a doubly-filtered homotopy type.

The topological link type determines the minimal plumbing resolution topology.

Abstract

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice homology is an invariant of the smooth knot type of a generalized algebraic knot in a rational homology sphere. In that case, knot lattice homology can be realized as the cellular homology of a doubly-filtered homotopy type, which is itself invariant. Along the way, we show that the topological link type of a generalized algebraic link determines the topology of the minimal plumbing resolution for the nested singularity type used to create it. Knot lattice homotopy is a natural invariant in that diffeomorphisms of the knot that play suitably well with the minimal good resolution will provide a contractible space of morphisms between the doubly-filtered knot lattice spaces associated to any presentation.