The Kauffman Bracket as an Evaluation of the Tutte

Polynomial

Whitney ShermanSaint Michael’s College

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What is a knot?

• A piece of string with a knot tied in it• Glue the ends together

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Movement

• If you deform the knot it doesn’t change.

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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.

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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.

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Links

• A set of knots, all tangled.

• The classic Hopf Links with two components and 10 components.

• The Borremean Rings with three components.

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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.

-+

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Links to Planar Graphs

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Kauffman Bracket in Terms of Pictures

• Three Rules – 1. – 2. a

b

– 3.

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The Connection

• Find the Kauffman Bracket values of and in the Tutte polynomial.0x 0y

=A< > + A < >

-1

=A(-A -A ) + A (1) = -A-22 3-1

=A< > + A < >-1

=A(1) + A (-A –A ) = -A-1 -22 -3

0x

0y

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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

1 /G A G e A G e

G e /G e

e

Ge

• positive crossing:– There exists a graph G such that 1/G A G e A G e

G

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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

f0x

0y

2 ( ) ( ) 2 4 4( ; , )V G E GL A T G A A

LG

( ) ( ) ( / )f G af G e bf G e ( ) ( ) ( )f GH f G f H

( ) ( ) 0 0( ) ( ; , )E r E r E x yf G a b T G

b a

GH G H G H

f

f

f

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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

f

( )f G G Aa

b 1A

1 /G A G e A G e

( ) ( ) ( / )f G af G e bf G e

( ) 1( ( )) 0 01

( ; , )E r E r E x yG A A T G

A A

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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

( ) ( ) ( )r A V G k A

( ) 1 1( ( ) 1 ) 0 01

( ; , )E V G V G x yG A A T G

A A

=2 ( ) 2 0 0

1( ; , )E V G x y

G A T GA A

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Final Touches

• With the values and

• Showing that the Kauffman bracket is an invariant of the Tutte polynomial.

2 ( ) 2 4 4( ; , )E V GG A T G A A

30x A 3

0y A

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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.

1L 2L 1L

2L

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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