Linking number Smallest number of crossing changes that make a link splittable Lk(K)=Tw(K)+Wr(K) Analogous to unknotting no. eqn used in DNA research…

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Presentation transcript:

Linking number Smallest number of crossing changes that make a link splittable Lk(K)=Tw(K)+Wr(K) Analogous to unknotting no. eqn used in DNA research… defines linking no for a knot. Stronger invariant

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

Plane homeo to hollow box

Knot Polynomials ways to represent a knot algebraically Bracket and Jones polynomials Conway, Alexander, HOMFLY polynomials Kauffman polynomial Alexander is the oldest. Kauffman c. 1985. Bracket has alaogy to statmech partition function

Knot Algebras Knot Groups Quaternion groups and the "Type IV Redermeister move" Tensor calculus using knots Not much detail… quite intimidating… same processes different notations and applications Commutative and non-commutative operations

Applications Biology-the knotting of DNA strands Physics Conductance Yang-Baxter equations The Lorentz attractor as a knot Atoms were postulated to be knotted vortices [KP,5]. Knotting-unknotting may explain how enzymes work in the body

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 KnotPlot Site. http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html. Oct. 2000. Real Knots: Knotting, bends and hitches. http://www.RealKnots.com. May 2000. Ideas, Concepts, and Definitions. http://www.c3.lanl.gov/mega-math/gloss/knots/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.

KnotPlot Demonstration Zoo and relaxations