1. Spin Models and Distance-Regular Graphs

2. Spin Models and Distance-Regular Graphs

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

4. 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

5. 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

6. 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…

7. 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

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

9. 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

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

11. 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.

12. 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

13. 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

14. 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…

15. 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:

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

17. Spin Models and Link Invariance
Reidemeister Move I

18. Spin Models and Link Invariance
Reidemeister Move II

19. Spin Models and Link Invariance
Reidemeister Move III _

20. 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.

21. 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 :

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

23. 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

24. 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

25. 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

26. Spin Models and Link Invariance
Example.

27. Spin Models and Link Invariance

28. 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 …

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

30. Spin Models and Link Invariance
Reidemeister Moves II and III _

31. 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

32. 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

33. Association Schemes + Bose-Mesner Algebras

34. 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

35. 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

36. 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

37. 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 …

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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?