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An Untangled Introduction to Knot Theory Ana Nora Evans University of Virginia Mathematics 12 February 2010 Slide 2 Why study knots? Used in Biology Chemistry Physics Graph Theory .. 2 Rule Asia Slide 3 Gordian Knot Seeing Gordius, therefore, the people made him king. In gratitude, Gordius dedicated his ox cart to Zeus, tying it up with a highly intricate knot - - the Gordian knot. Another oracle -- or maybe the same one, the legend is not specific, but oracles are plentiful in Greek mythology -- foretold that the person who untied the knot would rule all of Asia. From Untying the Gordian Knot by Keith Devlin ( 3 Slide 4 Problem Solved!!! Alexander cuts the Gordian Knot, by Jean-Simon Berthlemy 4 Slide 5 More importantly Its fun Interesting Hot area of research 5 The Knot Book by Colin C. Adams An Introduction to Knot Theory by W.B. Raymond Lickorish Slide 6 Knot Definition 1.Take a piece a string. 2.Tie a knot in it. 3.Now glue the ends of the string together to form a knotted loop. (see Colin Adams The Knot Book) A knot is an embedding of S 1 in R 3. 6 Slide 7 Unknot 7 Slide 8 Trefoil 8 Slide 9 Figure 8 9 Slide 10 Which knot is this? 10 Slide 11 Planar Diagrams 11 Slide 12 Examples 12 Slide 13 Knot equivalence Two knots are equivalent if you can get one from the other by deforming the string without cutting it. Two knots are equivalent if there exists an isotopy of R 3 taking one knot to the other. 13 Slide 14 Reidemeister Move 1 (R1) 14 Slide 15 Reidemeister Move 2 (R2) 15 Slide 16 Reidemeister Move 3 (R3) 16 Slide 17 Are these three moves enough? 17 Slide 18 More Moves 18 Slide 19 Which knot is this? 19 Slide 20 Knot Invariants A knot invariant is a mathematical object associated to a knot two equivalent knots have the same invariant Examples Crossing number Unknotting number 20 Slide 21 James Waddell Alexander II Topological invariants of knots and links In Transactions of the American Mathematical Society (1928) 21 Slide 22 John Conway An enumeration of knots and links, and some of their algebraic properties In Computational Problems in Abstract Algebra, 1967 22 Slide 23 Oriented Knots Positive crossing Negative crossing 23 Slide 24 Conways Skein Relation 24 Slide 25 Example - Trefoil 25 Slide 26 Example Hopf Link 26 Slide 27 Example Unlink 27 Slide 28 Example Unlink 28 Slide 29 Back to Trefoil 29 Slide 30 Vaughan Jones Fields Medal in 1990 A polynomial invariant for knots via von Neumann algebras. In Bull. Amer. Math. Soc. (N.S.) Volume 12, Number 1 (1985), 30 Slide 31 Kauffman Bracket 31 Slide 32 Kauffman Bracket - example 32 Slide 33 Jones Polynomial 33 Slide 34 Kauffman Bracket R2 34 Slide 35 Jones Polynomial R2 35 Slide 36 Trefoil 36 Slide 37 From: On Khovanov's Categorification of the Jones Polynomial , Dror Bar-Natan, Algebraic and Geometric Topology 2-16 (2002) 337-370 37 Slide 38 Open Question Is there a nontrivial knot with Jones polynomial equal to that of the unknot? It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite. 38 Slide 39 Thank you! The talk is available at: 39