An Untangled Introduction to Knot Theory

Ana Nora EvansUniversity of Virginia Mathematics

12 February 2010

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Why study knots?

• Used in– Biology– Chemistry– Physics– Graph Theory– …..

Rule Asia

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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)

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Problem Solved!!!

Alexander cuts the Gordian Knot, by Jean-Simon Berthélemy

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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

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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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Unknot

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Trefoil

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Figure 8

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Which knot is this?

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Planar Diagrams

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Examples

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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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Reidemeister Move 1 (R1)

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Reidemeister Move 2 (R2)

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Reidemeister Move 3 (R3)

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Are these three moves enough?

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More Moves

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Which knot is this?

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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

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James Waddell Alexander II

“Topological invariants of knots and links”

In Transactions of the American Mathematical Society (1928)

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John Conway

“An enumeration of knots and links, and some of their algebraic properties”

In Computational Problems in Abstract Algebra, 1967

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Oriented Knots

Positive crossing

Negative crossing

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Conway’s Skein Relation

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Example - Trefoil

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Example – Hopf Link

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Example – Unlink

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Example – Unlink

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Back to Trefoil

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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),

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Kauffman Bracket

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Kauffman Bracket - example

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Jones Polynomial

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Kauffman Bracket – R2

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Jones Polynomial – R2

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Trefoil

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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.

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Thank you!

The talk is available at:

http://www.cs.virginia.edu/nora