Derivatives with Alexander pairs for quandles
Volume 259 / 2022
Fundamenta Mathematicae 259 (2022), 1-31
MSC: Primary 57K12; Secondary 57K10, 57K14.
DOI: 10.4064/fm890-12-2021
Published online: 18 June 2022
Abstract
Twisted Alexander invariants for finitely presented groups equipped with a linear representation are defined using free derivatives introduced by R. H. Fox. In this paper, we define derivatives for quandles and use them to introduce new invariants for finitely presented quandles equipped with a quandle representation. We show that twisted Alexander invariants and the quandle cocycle invariants can be obtained in our framework using suitable choices of augmented Alexander pairs. As an application, we define a new invariant of $n$-moves and show that it can be applied to distinguish $5$-move equivalence classes for some knots.
