Spin Models and Distance-Regular Graphs

Spin Models and Distance-Regular Graphs

Spin Models and Distance-Regular Graphs By John Caughman Joint work with Nadine Wolff

Outline of part I History Knots, links, and link diagrams Spin models and link invariants Association schemes and Bose-Mesner algebras Summary of part I

History 1990 – V. Jones wins Fields medal for work connecting statistical mechanics to link invariants Constructed a new link invariant from a matrix known as a spin model 1993 – F. Jaeger gives topological proof that spin models are contained in Bose-Mesner algebras 1995 – K. Nomura proves Jaeger’s result using linear algebra

Knots, links, and link diagrams Knot – piece-wise linear simple closed curve in Euclidean 3-space ℝ³ Link – finite union of pairwise disjoint knots Equivalent links – L1 ~ L2 if there exists an orientation-preserving homeomorphism of ℝ³ to itself mapping L1 onto L2 Link diagram – projection of link onto ℝ²  Spatial information can be lost…

Knots, links, and link diagrams The projection must satisfy: No three points on the link project to the same point in the diagram Only finitely many points in the diagram correspond to more than one point on the link  At each crossing point we indicate the spatial structure of the link

Knots, links, and link diagrams Different projections of figure-eight knot

Knots, links, and link diagrams Q: How can we tell whether two different link diagrams represent equivalent links? There are three operations, called Reidemeister moves, that can be applied to a link diagram without changing the link it represents

Reidemeister moves I, II, and III Knots, links, and link diagrams Reidemeister moves I, II, and III

Knots, links, and link diagrams Theorem. Link diagrams determine equivalent links if and only if one can be obtained from the other by a sequence of Reidemeister moves. Ref.: C. Livingston; Knot Theory.

Knots, links, and link diagrams To show two links are equivalent, must show there exists a finite sequence of Reidemeister moves changing one link diagram to the other Conversely, to show two links are not equivalent, we must show there does NOT exist such a sequence But … there is no known bound on the length of such a sequence, so an exhaustive search is not possible

Knots, links, and link diagrams Try to find properties of link diagrams that remain unchanged by Reidemeister moves These are called link invariants If a link invariant assumes different values for two given diagrams, then the diagrams represent different links

Knots, links, and link diagrams We call a property that is invariant only under RM moves II and III a partial link invariant Two link diagrams that differ only by RM moves II and III are called regularly isotopic To get our invariant we begin by shading the link diagram…

Spin Models and Link Invariance Two-color theorem: the regions - or ‘faces’ - of a link diagram can always be colored black or white so that adjacent regions are different ‘Checkerboard’ coloring: assumes the unbounded region is colored white Sign convention for crossings:

Spin Models and Link Invariance Construction of Tait graph: + _ _

Spin Models and Link Invariance Reidemeister Move I

Spin Models and Link Invariance Reidemeister Move II

Spin Models and Link Invariance Reidemeister Move III _

Spin Models and Link Invariance Definition. A spin model is a triple where , and are symmetric n x n matrices with entries in ℂ that satisfy the following equation: The elements of X are called the spins of S.

Spin Models and Link Invariance In order for a spin model to give rise to a partial link invariant, it must satisfy the following invariance equations ∀ a,b,c ∊ X :

Spin Models and Link Invariance If the Type II equation holds, then the Type III equation is equivalent to

Spin Models and Link Invariance Let b=c in the Type III equation: Let a=c in the Type III* equation: have constant row sums and constant diagonal entries

Spin Models and Link Invariance Define the modulus of S to be the diagonal entry of Example of a spin model with : Constant row sums 2i , - 2i Modulus = - i

Spin Models and Link Invariance Definition. spin model satisfying Type II, III equations. L is a link diagram and L L the Tait graph of L with vertices V. Let n = |X|. A state σ is a function from V to X. Then the partition function is defined to be

Spin Models and Link Invariance Example.

Spin Models and Link Invariance

Spin Models and Link Invariance Theorem. Let S be a spin model satisfying the Type II and III equations and let L1 and L2 be connected link diagrams. Then provided that L1 and L2 differ only by Reidemeister moves II and III. Idea of proof. We must show that for RM moves II and III the partition function remains invariant. Recall that RM moves separate into sub-cases depending on the shadings. To illustrate we demonstrate the computation for one case of RM move II …

Spin Models and Link Invariance Reidemeister Move II LL1 LL2 _ +

Spin Models and Link Invariance Reidemeister Moves II and III _

Spin Models and Link Invariance Theorem. Let S be a spin model satisfying the Type II and III equations and let L1 and L2 be connected link diagrams. Let m be the modulus of S. If L1 and L2 differ only by Reidemeister move I, then where p = ±1. Value of p depends only on type of crossing involved. Proof involves similar computations as before

Spin Models and Link Invariance The partition function is a partial link invariant Partition function invariant under Reidemeister moves II and III But the partition function behaves predictably under Reidemeister move I The partition function can be modified to give a link invariant

Association Schemes + Bose-Mesner Algebras

Association Schemes + Bose-Mesner Algebras M is commutative, since associate matrices are symmetric M is closed under entry-wise matrix product, called the Hadamard product, since each Ai is a 0-1 matrix

Association Schemes + Bose-Mesner Algebras Examples of symmetric association schemes and Bose-Mesner Algebras arise from distance-regular graphs Graphs formed by the edges and vertices of the 5 Platonic solids are examples of distance-regular graphs

Association Schemes + Bose-Mesner Algebras Consider the tetrahedron The associate matrices are the distance matrices A0, A1 satisfy the axioms of a symmetric association scheme A0, A1 form basis for a Bose-Mesner algebra called the adjacency algebra of the graph

Association Schemes + Bose-Mesner Algebras Our example for a spin model is an element of the adjacency algebra of the tetrahedron This fact holds in general …

Association Schemes + Bose-Mesner Algebras Theorem. If a spin model satisfies the Type II and III equations, then it naturally gives rise to a Bose-Mesner algebra for an association scheme, such that the matrices are elements of this algebra. Jaeger first proved this topologically, then a simpler proof was found by Nomura, using linear algebra

Association Schemes + Bose-Mesner Algebras Outline of proof. define n-dim. column vector ub,c with x-entry Define N to be the set of all symmetric n x n matrices A such that for all is an eigenvector of A. N is a subspace of Matx (ℂ) and contains identity matrix We want to show N is a Bose-Mesner algebra for some association scheme and contains

Association Schemes + Bose-Mesner Algebras Theorem. An algebra of symmetric n x n matrices is the Bose-Mesner algebra of some association scheme iff it contains the identity matrix I, the all 1’s matrix J, and is closed under the Hadamard product. Ref.: Brouwer, Cohen, Neumaier; Distance-Regular Graphs

Association Schemes + Bose-Mesner Algebras To show N a Bose-Mesner algebra we need to show N is closed under the ordinary matrix product N is closed under the Hadamard matrix product N contains J, the all 1’s matrix

Association Schemes + Bose-Mesner Algebras To show N contains , we need Type III equation Rewrite using definition of Similarly we can show N contains using the Type III* equation

Summary of Part One Spin models of Type I,II are contained in Bose-Mesner algebras of association schemes This narrows the search for spin models But…not all Bose-Mesner algebras of association schemes contain the matrices of spin models So which association schemes support spin models?