The Kauffman Bracket as an Evaluation of the Tutte Polynomial

1 / 17

# The Kauffman Bracket as an Evaluation of the Tutte Polynomial - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## The Kauffman Bracket as an Evaluation of the Tutte Polynomial

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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.
• 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
• 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.
• A set of knots, all tangled.
• The classic Hopf Links with two components and 10 components.
• The Borremean Rings with three components.
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.

+

-

=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
• 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
• 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
• 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 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
• With the values and
• Showing that the Kauffman bracket is an invariant of the Tutte polynomial.
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
• 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