Signatures of iterated torus links

TL;DR

This paper computes multivariate signatures of Seifert links, especially cored torus links, using Neumann's formulas and geometric interpretations, advancing the understanding of link signatures in 3-manifold topology.

Contribution

It extends the computation of multivariate signatures to Seifert and cored torus links, providing new formulas and geometric interpretations based on Neumann's work.

Findings

Computed signatures for cored torus links using Neumann's formulas.

Rewritten signature formulas in terms of integral points in parallelograms.

Connected signature computations to geometric lattice point counting methods.

Abstract

We compute the multivariate signatures of any Seifert link (that is a union of some fibers in a Seifert homology sphere), in particular, of the union of a torus link with one or both of its cores (cored torus link). The signatures of cored torus links are used in Degtyarev-Florens-Lecuona splicing formula for computation of multivariate signatures of cables over links. We use Neumann's computation of equivariant signatures of such links. For signatures of torus links with the core(s) we also rewrite the Neumann's formula in terms of integral points in a certain parallelogram, similar to Hirzebruch's formula for signatures of torus links (without cores) via integral points in a rectangle.