The Kauffman Bracket as an Evaluation of the Tutte Polynomial Whitney Sherman Saint Michael’s College

2 What is a knot? A piece of string with a knot tied in it Glue the ends together

3 Movement If you deform the knot it doesn’t change.

4 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.

5 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.

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

7 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. -+

8 Links to Planar Graphs

9 Kauffman Bracket in Terms of Pictures Three Rules –1. –2. a b –3.

10 The Connection Find the Kauffman Bracket values of and in the Tutte polynomial. =A + A =A(-A -A ) + A (1) = -A -223 =A + A =A(1) + A (-A –A ) = -A

11 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

12 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

13 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

14 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 =

15 Final Touches With the values and Showing that the Kauffman bracket is an invariant of the Tutte polynomial.

16 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.

17 Resources Pictures taken from – /scharein/KnotPlot.html Other information from –The Knot Book, Colin Adams –Complexity: Knots, Colourings and Counting, D. J. A. Welsh –Jo Ellis-Monaghan