An Untangled Introduction to Knot Theory Ana Nora Evans University of Virginia Mathematics 12...
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Transcript of An Untangled Introduction to Knot Theory Ana Nora Evans University of Virginia Mathematics 12...
An Untangled Introduction to Knot Theory
Ana Nora EvansUniversity of Virginia Mathematics
12 February 2010
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 (www.maa.org/devlin/devlin_9_01.html)
5
More importantly
• It’s fun• Interesting• Hot area of research
The Knot Book by Colin C. AdamsAn Introduction to Knot Theory by W.B. Raymond Lickorish
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 S1 in R3.
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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 R3 taking one knot to the other.
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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
21
James Waddell Alexander II
“Topological invariants of knots and links”
In Transactions of the American Mathematical Society (1928)
22
John Conway
“An enumeration of knots and links, and some of their algebraic properties”
In Computational Problems in Abstract Algebra, 1967
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),
37
From: “On Khovanov's Categorification of the Jones Polynomial “, Dror Bar-Natan,Algebraic and Geometric Topology 2-16 (2002) 337-370
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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.