Lots About Knots With a View Towards the Jones Polynomial Anne-Marie Oreskovich Math 495-B, Spring 2000

Examples of Knots Trefoil Knot Figure-Eight Knot Unknot

Knots in History Ancient Cave Drawings Weaving, Bridgebuilding, Sailing Basis of Mayan Mathematical System Mythology

What is Knot Theory? Knot Theory is the branch of mathematics devoted to studying knots.

What then, pray tell, is a knot? A knot is a closed curve in space that doesn’t intersect itself anywhere.

But how can we tell two knots apart?

POLYNOMIALS! Any two projections of the same knot yield the same polynomial (true for all types of polynomials people have discovered for knots…Jones, Alexander, HOMFLY, bracket, etc.) If a given type of polynomial is different for two knots, then the knots are distinct

Alexander Polynomial Step 1: Orient the knot K and choose a crossing: Step 2: Compute A(K + ), A(K - ), and A(K 0 ), where K + is the knot with circled crossing looking like, K - is the knot with circled crossing looking like, and K 0 is the knot with circled crossing looking like

Step 3: Calculate A of the entire knot K using the following 2 rules: a) A( )=0 and b) A(K + )-A(K - )+(t 1/2 -t -1/2 )A(K 0 )=0.

K+K+ K_ KoKo

Rule 2: A(K + )-A(K - )+(t 1/2 -t -1/2 )A(K 0 )=0 A( )-A( )+(t 1/2 -t -1/2 )A( )=0 But we just saw in the last slide that = (and therefore has Alexander polynomial equal to 1 by Rule 1), so we just need to solve for A( ).

A( )-A( )+(t 1/2 -t -1/2 )A( )=0 but A( )=0 ( because A of any splittable link is 0), so A( )=-t 1/2 +t -1/2 Hence A( )=(t 1/2 -t -1/2 ) 2 +1= t+t

Conclusions Knots are cool Knots are fun One way to tell two knots apart is by computing their polynomials The Alexander polynomial is one such example, computed by choosing a crossing of the knot, computing the Alexander polynomial of the knot with three different variations on the crossing, and using a formula to relate all three subsequent Alexander polynomials

References C. Adams, The Knot Book, W. H. Freeman and Company, New York (1994) N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford UP, New York (1994) “A calculus student came in for help, and after the teacher had worked some problems, the student said,“So what kind of math do you like?” The teacher said, “Knot theory.” The student said, “Yeah, me either.” (from Marty Scharlemann, told in The Knot Book)