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The Kauffman Bracket as an Evaluation of the Tutte Polynomial. Whitney Sherman Saint Michael’s College. What is a knot?. A piece of string with a knot tied in it Glue the ends together. Movement. If you deform the knot it doesn’t change. The Unknot. The simplest knot.

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the kauffman bracket as an evaluation of the tutte polynomial

The Kauffman Bracket as an Evaluation of the Tutte Polynomial

Whitney Sherman

Saint Michael’s College

what is a knot
What is a knot?
  • A piece of string with a knot tied in it
  • Glue the ends together
movement
Movement
  • If you deform the knot it doesn’t change.
the unknot
The Unknot
  • The simplest knot.
  • An unknotted circle, or the trivial knot.
  • You can move from the one view of a knot to another view using Reidemeister moves.
reidemeister moves
Reidemeister Moves
  • First: Allows us to put in/take out a twist.
  • Second: Allows us to either add two crossings or remove two crossings.
  • Third: Allows us to slide a strand of the knot from one side of a crossing to the other.
links
Links
  • A set of knots, all tangled.
  • The classic Hopf Links with two components and 10 components.
  • The Borremean Rings with three components.
labeling technique
Labeling Technique

Begin with the shaded knot projection.

  • If the top strand ‘spins’ left to sweep out black then it’s a + crossing.
  • If the top strand ‘spins’ right then it’s a – crossing.

+

-

the connection

=A< > + A < >

-1

=A(-A -A ) + A (1) = -A

2

-2

-1

3

=A< > + A < >

-1

2

-3

=A(1) + A (-A –A ) = -A

-1

-2

The Connection
  • Find the Kauffman Bracket values of and in the Tutte polynomial.
kauffman bracket in polynomial terms
Kauffman Bracket In Polynomial Terms
  • if is an edge corresponding to:
  • negative crossing:
    • There exists a graph such that where and denote deletion and contraction of the edge from
  • positive crossing:
    • There exists a graph G such that
recall universality property
Recall Universality Property
  • Some function on graphs such that and

(where is either the disjoint union of and or where and share at most one vertex)

  • is given by value takes on bridges

value takes on loops

Tutte polynomial

  • The Universality of the Tutte Polynomial says that any invariant which satisfies those two properties is an evaluation of the Tutte polynomial
  • If is an alternating positive link diagram then the Bracket polynomial of the unsigned graph is
the connection cont
The Connection Cont
  • We know from the Kauffman Bracket that , and from that
  • By replacing with , with , and

with … we get one polynomial from the other.

  • With those replacements the function becomes
the connection cont14
The Connection Cont
  • Recall: Rank by definition is the vertex set minus the number of components of the graph (which in our case is 1)
  • With those replacements

=

final touches
Final Touches
  • With the values and
  • Showing that the Kauffman bracket is an invariant of the Tutte polynomial.
applications of the kauffman bracket
Applications of the Kauffman Bracket
  • It is hard to tell unknot from a messy projection of it, or for that matter, any knot from a messy projection of it.
  • If does not equal , then can’t be the same knot as . 
  • However, the converse is not necessarily true.
resources
Resources
  • Pictures taken from
    • http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html
  • Other information from
    • The Knot Book, Colin Adams
    • Complexity: Knots, Colourings and Counting, D. J. A. Welsh
    • Jo Ellis-Monaghan