Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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C. Collari - P. Lisca

Symmetric union diagrams and refined spin models

created by collari on 18 Sep 2018
modified on 21 May 2019


Published Paper

Inserted: 18 sep 2018
Last Updated: 21 may 2019

Journal: Canadian Mathematical Bulletin
Year: 2018
Doi: doi:10.4153/S0008439518000115

ArXiv: 1804.09157 PDF


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 inequivalent 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 inequivalent 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.

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