Abelian link invariants and homology

TL;DR

This paper explores abelian link invariants derived from quantum Chern-Simons theory, examining their relation to homology groups of link complements, and demonstrating that certain invariants depend on more than just homology or homotopy type.

Contribution

It establishes the connection between abelian link invariants and homology groups, and shows the U(1) Reshetikhin-Turaev invariant's dependence beyond homology or homotopy.

Findings

Invariants coincide with those of S^3 under certain conditions.

U(1) Reshetikhin-Turaev invariant depends on more than homology or homotopy.

Relation between abelian invariants and the homology of link complements.

Abstract

We consider the link invariants defined by the quantum Chern-Simons field theory with compact gauge group U(1) in a closed oriented 3-manifold M. The relation of the abelian link invariants with the homology group of the complement of the links is discussed. We prove that, when M is a homology sphere or when a link -in a generic manifold M- is homologically trivial, the associated observables coincide with the observables of the sphere S^3. Finally we show that the U(1) Reshetikhin-Turaev surgery invariant of the manifold M is not a function of the homology group only, nor a function of the homotopy type of M alone.