Lattices, graphs, and Conway mutation

TL;DR

This paper establishes a connection between lattice invariants and graph isomorphism, showing that certain link diagrams are uniquely determined by topological invariants, with implications for knot mutation classification.

Contribution

It proves that the d-invariant of a lattice of integral cuts determines a graph's 2-isomorphism type and links this to mutation invariants of alternating links via Heegaard Floer homology.

Findings

The 2-isomorphism type of a connected graph is determined by the d-invariant.

Alternating links with homeomorphic branched double-covers are mutants.

Heegaard Floer homology classifies certain link mutations.

Abstract

The d-invariant of an integral, positive definite lattice L records the minimal norm of a characteristic covector in each equivalence class mod 2L. We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral cuts (or flows). As an application, we prove that a reduced, alternating link diagram is determined up to mutation by the Heegaard Floer homology of the link's branched double-cover. Thus, alternating links with homeomorphic branched double-covers are mutants.