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Symmetric Union Diagrams and Refined Spin Models

Published online by Cambridge University Press:  09 November 2018

Carlo Collari
Mathematical Sciences, Durham University, UK Email:
Paolo Lisca
Department of Mathematics, University of Pisa, Italy Email:


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.

© Canadian Mathematical Society 2018 

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


de la Harpe, Pierre, Spin models for link polynomials, strongly regular graphs and Jaeger’s Higman-Sims model . Pacific J. Math. 162(1994), no. 1, 5796.Google Scholar
Eisermann, Michael and Lamm, Christoph, Equivalence of symmetric union diagrams . J. Knot Theory Ramifications 16(2007), no. 07, 879898. Scholar
Eisermann, Michael and Lamm, Christoph, A refined Jones polynomial for symmetric unions . Osaka J. Math. 48(2011), no. 2, 333370.Google Scholar
Goldschmidt, David M. and Jones, Vaughan F. R., Metaplectic link invariants . Geom. Dedicata 31(1989), no. 2, 165191. Scholar
Jaeger, François, Matsumoto, Makoto, and Nomura, Kazumasa, Bose-Mesner Algebras Related to Type II Matrices and Spin Models . J. Algebraic Combin. 8(1998), no. 1, 3972. Scholar
Jones, Vaughan, On knot invariants related to some statistical mechanical models . Pacific J. Math. 137(1989), no. 2, 311334.Google Scholar
Jones, Vaughan F. R., On a certain value of the Kauffman polynomial . Commun. Math. Phys. 125(1989), no. 3, 459467.Google Scholar
Nomura, Kazumasa, An algebra associated with a spin model . J. Algebraic Combin. 6(1997), no. 1, 5358. Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.0), Scholar