On colored Turaev-Viro invariants for links in arbitrary 3-manifolds

TL;DR

This paper introduces colored Turaev-Viro invariants for links in 3-manifolds, exploring their properties, differences from classical invariants, and relationships with link complements, providing new insights into link invariants.

Contribution

It defines and analyzes colored Turaev-Viro invariants for links in 3-manifolds, highlighting their properties and distinguishing features from classical polynomial invariants.

Findings

Colored Turaev-Viro invariants can distinguish links with identical HOMFLY and Kauffman polynomials.

These invariants are sometimes but not always determined by the invariants of link complements.

The invariants behave predictably under connected sums of pairs and along links.

Abstract

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in the three-sphere having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev-Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev-Viro invariants of (M,L) and those of the complement of L in M, showing that they are sometimes but not always determined by each other.