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Symmetric Union Diagrams and Refined Spin Models

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 November 2018

Carlo Collari  and
Paolo Lisca
Carlo Collari
Affiliation:
Mathematical Sciences, Durham University, UK Email: carlo.collari.math@gmail.com
Paolo Lisca
Affiliation:
Department of Mathematics, University of Pisa, Italy Email: paolo.lisca@unipi.it
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Abstract

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An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 451 - 468
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The first author was partially supported by an Indam grant, and hosted by the IMT in Toulouse, during the early stages of this paper.

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