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Slopes and signatures of links

Volume 258 / 2022

Alex Degtyarev, Vincent Florens, Ana G. Lecuona Fundamenta Mathematicae 258 (2022), 65-114 MSC: Primary 57M27. DOI: 10.4064/fm136-1-2022 Published online: 14 March 2022


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.


  • Alex DegtyarevDepartment of Mathematics
    Bilkent University
    06800 Ankara, Turkey
  • Vincent FlorensLaboratoire de Mathématiques
    et de leurs Applications
    UMR CNRS 5142
    Université de Pau
    et des Pays de l’Adour
    Avenue de l’Université, BP 1155
    64013 Pau Cedex, France
  • Ana G. LecuonaSchool of Mathematics and Statistics
    University of Glasgow
    University Place
    Glasgow, G12 8QQ, UK

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