Spin Models and Distance-Regular Graphs

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Spin Models and Distance-Regular Graphs
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Spin Models and Distance-Regular Graphs
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Spin Models and Distance-Regular Graphs

• By John Caughman

Joint work with Nadine Wolff
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Outline of part I

1. History
2. Knots, links, and link diagrams
3. Spin models and link invariants
4. Association schemes and Bose-Mesner algebras
5. Summary of part I

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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 Jaegers result using
  linear algebra

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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
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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

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Knots, links, and link diagrams

• Different projections of figure-eight knot

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

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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

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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

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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

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Spin Models and Link Invariance
Construction of Tait graph
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Spin Models and Link Invariance
Construction of Tait graph
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Spin Models and Link Invariance
Reidemeister Move I
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Spin Models and Link Invariance
Reidemeister Move II
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Spin Models and Link Invariance
Reidemeister Move III
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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.
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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
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Spin Models and Link Invariance
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Spin Models and Link Invariance
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Spin Models and Link Invariance

• Define the modulus of S to be the diagonal entry
  of

• Constant row sums 2i , - 2i
• Modulus - i

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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 s is a function from V to X. Then the
partition function is defined to be
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Spin Models and Link Invariance
Example.
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Spin Models and Link Invariance
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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

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Spin Models and Link Invariance
Reidemeister Move II
LL1
LL2
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Spin Models and Link Invariance
Reidemeister Moves II and III
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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

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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

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Association Schemes Bose-Mesner Algebras
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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

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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

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Association Schemes Bose-Mesner Algebras
Consider the tetrahedron

• 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

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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
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Association Schemes Bose-Mesner Algebras

• Jaeger first proved this topologically, then a
  simpler proof was found by Nomura, using linear
  algebra

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Association Schemes Bose-Mesner Algebras
Outline of proof.

• N is a subspace of Matx (C) and contains
  identity matrix

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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 1s matrix J, and is closed under the
Hadamard product. Ref. Brouwer, Cohen,
Neumaier Distance-Regular Graphs
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Association Schemes Bose-Mesner Algebras
To show N a Bose-Mesner algebra we need to show
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Association Schemes Bose-Mesner Algebras
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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
• Butnot all Bose-Mesner algebras of association
  schemes contain the matrices of spin models
• So which association schemes support spin models?

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Thank You Paul Terwilliger Nadine Wolff
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THE END