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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?. 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

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  1. 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. =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.

  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 • 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

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