Slopes and signatures of links
Volume 258 / 2022
Fundamenta Mathematicae 258 (2022), 65-114
MSC: Primary 57M27.
DOI: 10.4064/fm136-1-2022
Published online: 14 March 2022
Abstract
We define the slope of a colored link in an integral homology sphere, associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima–Yamasaki $\eta $-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. The slope is responsible for an extra correction term in the signature formula for the splice of two links, in the previously open exceptional case where both characters are admissible. Using a similar construction for a special class of tangles, we formulate generalized skein relations for the signature.
