NDLI: Symmetric Versus Non-Symmetric Spin Models for Link Invariants

Content Provider	Springer Nature Link
Author	Jaeger, François mura, Kazumasa
Copyright Year	1999
Abstract	We study spin models as introduced in [20]. Such a spin model can be defined as a square matrix satisfying certain equations, and can be used to compute an associated link invariant. The link invariant associated with a symmetric spin model depends only trivially on link orientation. This property also holds for quasi-symmetric spin models, which are obtained from symmetric spin models by certain “gauge transformations” preserving the associated link invariant. Using a recent result of [16] which asserts that every spin model belongs to some Bose-Mesner algebra with duality, we show that the transposition of a spin model can be realized by a permutation of rows. We call the order of this permutation the index of the spin model. We show that spin models of odd index are quasi-symmetric. Next, we give a general form for spin models of index 2 which implies that they are associated with a certain class of symmetric spin models. The symmetric Hadamard spin models of [21] belong to this class and this leads to the introduction of non-symmetric Hadamard spin models. These spin models give the first known example where the associated link invariant depends non-trivially on link orientation. We show that a non-symmetric Hadamard spin model belongs to a certain triply regular Bose-Mesner algebra of dimension 5 with duality, and we use this to give an explicit formula for the associated link invariant involving the Jones polynomial.
Starting Page	241
Ending Page	278
Page Count	38
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	10
Issue Number	3
e-ISSN	15729192
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	1999-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computer Science Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Combinatorics Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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