Linking number Smallest number of crossing changes that

Linking number • Smallest number of crossing changes that make a link splittable • Lk(K)=Tw(K)+Wr(K)

Seifert Spanning Surfaces • One of the stronger knot invariants • Looking at the space “covered” by a knot • Preserved under homeomorphism

Knot Polynomials • ways to represent a knot algebraically • Bracket and Jones polynomials • Conway, Alexander, HOMFLY polynomials • Kauffman polynomial

Knot Algebras • Knot Groups • Quaternion groups and the "Type IV Redermeister move" • Tensor calculus using knots

Applications • Biology-the knotting of DNA strands • Physics – Conductance – Yang-Baxter equations • The Lorentz attractor as a knot

Summary • Recap of Knot Theory • Why Knot Theory?

Catalog of Knots and Knot Nomenclature • Arbitrary system from late 1800’s • Apparently classified by number of crossings

Knots On the Web • Scharein, Robert G. The Knot. Plot Site. http: //www. cs. ubc. ca/nest/imager/contributions/s charein/Knot. Plot. html. Oct. 2000. • Real Knots: Knotting, bends and hitches. http: //www. Real. Knots. com. May 2000. • Ideas, Concepts, and Definitions. http: //www. c 3. lanl. gov/megamath/gloss/knots. html

Bibliography • Adams, Colin. The Knot Book. New York: W. H Freeman and Co. , 1994 • Farmer, David W. and Theodore B. Stanford. Knots and Surfaces. American Mathematical Society, 1996. • Gilbert, N. D. and T. Porter. Knots and Surfaces. New York: Oxford University Press, 1994. • Kauffman, Louis. Knots and Physics, Second Ed. New Jersey: World Scientific, 1993. On Knots. New Jersey: Princeton University Press, 1987. • Prasilov, V. V. Intuitive Topology. American Mathematical Society, 1995.

Knot. Plot Demonstration