By Nadine Wolff

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Spin Models and Bose-Mesner Algebras

• By Nadine Wolff

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Outline

• History
• Links and link diagrams
• Spin models and link invariance
• Association schemes and Bose-Mesner algebras
• Conclusion

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History

• 1990 V. Jones wins Fields medal for work
  connecting statistical mechanics to link
  invariants
• Constructs link invariant from spin models
• 1993 F. Jaeger proves spin models are contained
  in Bose-Mesner algebras using topology
• 1995 K. Nomura proves Jaegers result using
  linear algebra

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Links and Link Diagrams
Knot piece-wise linear simple closed curve in
Euclidean 3-space Link a finite collection of
pair-wise disjoint knots Equivalent links if
there exists orientation-preserving homeomorphism
R³ U 8 to itself mapping L1 onto L2 Link diagram
projection of link onto the plane ? Spatial
information can be lost
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Links and Link Diagrams

• We require the projection to 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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Links and Link Diagrams

• Different projections of figure-eight knot

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Links and Link Diagrams
Q How do we determine whether two different
link diagrams determine 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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Links and Link Diagrams
Reidemeister moves I, II, and III
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Links and Link Diagrams
Theorem. Two 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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Links and Link Diagrams

• To show two links are equivalent, must show there
  exists a sequence of RM 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, thus an exhaustive search is not
  possible

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Links and Link Diagrams

• Try to find properties of all link diagrams of a
  link that remain unchanged by Reidemeister moves
• We call such properties link invariants
• If a link invariant takes on different values for
  two link diagrams, then they represent different
  links

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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 regions of link diagram can be
  colored black and white s.t. adjacent regions are
  different
• Checkerboard coloring color unbounded region
  white
• Sign convention

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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 if 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 only a partial link
  invariant for unoriented links

• Partition function invariant under Reidemeister
  moves II and III

• Partition function behaves predictably under
  Reidemeister move I

• The partition function can be modified to give a
  link invariant for oriented links

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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 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

• We give an outline of Nomuras proof, which uses
  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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Conclusion

• The matrices of spin models satisfying Type II,
  III equations 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
• In fact, the classification of spin models is
  still open and an active area of research

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Thank You John Caughman Gerardo
Lafferriere Steve Bleiler John Krussel
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THE END