On Colored Turaev–Viro Invariants for Links in Arbitrary 3-Manifolds

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 $L\subset M$ is an oriented link, 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 $\mathbb S^3$ 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 $M\backslash L$, showing that they are sometimes, but not always, determined by each other, and discuss some relations with the Witten–Reshetikhin–Turaev invariants and the Jones polynomial.