Strongly regular graphs and spin models for the Kauffman polynomial
type: Item
instance of: scholarly article
title: Strongly regular graphs and spin models for the Kauffman polynomial
zbMATH Open document ID: 0773.57005
DOI: 10.1007/BF00147743
author: François Jaeger
published in: Geometriae Dedicata
publication date: 1993-04-01T00:00:00Z
review text: The author obtains specializations of the Kauffman polynomial for links, depending on the choice of a formally self-dual strongly regular graph with strongly regular subconstituents, as studied by \textit{P. J. Cameron}, \textit{J. M. Goethals} and \textit{J. J. Seidel} [J. Algebra 55, 257-280 (1978; Zbl 0444.05045)]. These graphs are rather scarce, and most of the known ones lead to known specializations of the Kauffman polynomial. The Higman-Sims graph however gives a new example. The connection between the graphs and the link invariants is obtained via spin models. The adjacency matrix of the given graph generates a 3- dimensional algebra which is closed under the componentwise Hadamard product, a so-called Bose-Meisner algebra, with suitable properties. Those properties make it possible to re-interpret the algebra as coming from a different type of generating matrix, which may be combinatorially interpreted as a spin model. This in turn yields a direct combinatorial definition of a link invariant. In order to establish this connection and, in particular, to single out the right classes of Bose-Meisner algebras and of graphs, a systematic study of the relationship between spin models and association schemes is given.
reviewed by: Rainer Löwen
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 82B23
zbMATH DE Number: 147286
zbMATH Keywords: specializations of the Kauffman polynomial for links
zbMATH Keywords: formally self-dual strongly regular graph
zbMATH Keywords: strongly regular subconstituents
zbMATH Keywords: Higman-Sims graph
zbMATH Keywords: spin models
zbMATH Keywords: Bose-Meisner algebra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147743
OpenAlex ID: W2078419515
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description: scientific article; zbMATH DE number 147286