Twisted acyclicity of a circle and link signatures

TL;DR

This paper extends the concept of link signatures to arbitrary oriented codimension two submanifolds in odd-dimensional spheres, allowing for transversal intersections and parametrization by the entire torus, with new inequalities.

Contribution

It introduces a novel generalization of Murasugi-Tristram signatures to intersecting submanifolds, broadening their applicability in link theory.

Findings

Extended signatures to intersecting submanifolds

Generalized Murasugi-Tristram inequalities

Parametrization by the full torus

Abstract

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries concerning links of codimension two. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi-Tristram inequalities and their generalizations are also extended to this setup.