Slopes and signatures of links

Date
2022-03-14
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Source Title
Fundamenta Mathematicae
Print ISSN
Electronic ISSN
1730-6329
Publisher
Instytut Matematyczny,Polish Academy of Sciences, Institute of Mathematics
Volume
258
Issue
2022
Pages
65 - 114
Language
English
Type
Article
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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 η-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.

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