Knot theory, geometry and more...

A mini-course at UBC Vancouver for Two Weeks in Vancouver

A Summer School for Women in Math

This is a 16.5 hs course prepared for the Summer School for Women in Math at UBC, given during August 2016. The overall aim is to introduce students to basic ideas in knot theory as well as the relation of this area with Geometry, as well as other areas as Topology, Physics, Biology, Chemistry, Art, etc. The days are organized as follows, and the lecture notes appear bellow:  

Lecture 1: An introduction to knots and links.

Lecture 2: Linking numbers and diagrams.

Lecture 3: Knots and fundamental groups.

Lecture 4: Knots and polynomials: Conway.

Lecture 5: Unknotting sequences.

Lecture 6: Braids and the braid group.

Lecture 7: Knots and polynomials: Jones.

Lecture 8: Knots and statistical mechanics.

Lecture 9: Knots and moduli spaces.

Lecture 10: Knots, physics and biology: cable tangling and protein folding.

Lecture 11: Knots, art and dance: digitalizing images and braid dancing.

The presentations for Lecture 10 and Lecture 11 can be seen here:

Knot theory and biological classifications

Knot theory and proteins by E. Carson and M. Yang

Knot theory and physics by N. Anikeeva, E. Korfanty, C. Li and R. Stiyer

Knot theory and dance by J. Chan, B. Lei, B. Tian, S. Wu

Knot theory and art

Here are some resources both on knot theory in general as well as on the topics we have touched during the week (of course, there are many many more and I’d encourage you to search for them if you like any subject in particular - or email me for more references). With * we have marked the papers that are chosen to be presented during Lectures 10 and 11.

Knot theory

The knot book by Collin Adams

Knot theory by Grant Walker

Knots and Art

Knots as Processes in Art and Mathematics by Bojana Ginn

Knots and art by Piotr Pieranski

Knots in Art by Slavik Jablan, Ljiljana Radovic, Radmila Sazdanovic and Ana Zekovic

  1. *Exploring Braids through Dance:The “Waves of Tory” Problem by Andrea Hawksley

  2. *Figurative Torus and Braids by Robert Bosch and Tom Wexler

Knots and Physics

The geometry of physics and knots by Sir Michael Atiyah

  1. *Spontaneous knotting of an agitated string by Dorian M. Raymer, Douglas E. Smith

Knots and physics by Louis Kauffman

Knot theory and physics by Louis Kauffman

Knots and Biology

  1. *New biologically motivated knot table by Brasher, Scharein, Vazquez

  2. *Subknots in ideal knots, random knots, and knotted proteins by Rawdon, Millett, Stasiak

Knots and Geometry

Singularities by Alan H. Durfee

Hyperbolic Knot Theory by Jessica S. Purcell