Place: MC5045

Instructor: Louis H. Kauffman

Office: MC 5132

Phone: (519) 888-4567 x5596

E-mail: kauffman@uic.edu

Web page: http://www.math.uic.edu/~kauffman

This page contains links to files useful for the course.

See Course Description -- pdf Download for a course description.

There are a number of papers available from this website that are either notes about knot theory or explanations of research that are accessible to this course. I encourage you to browse around on the site and on the world wide web as well.

It sometimes happens that pdf files print out upside down. Please note that there is a button on the Adobe Acrobat viewer that allows you to rotate all the pages by 90 or 180 degrees.

See Braids and Quantum Gates -- pdf Download for a new paper on relationships between topology and quantum information theory. This paper contains an introduction to the Yang-Baxter Equation, Braid group, Markov Theorem, the construction of a new invariant of knots and links and applications to quantum computing.

See Knots and Hopf Algebras -- pdf Download for a paper on invariants of knots and links derived from Hopf algebras. Please read this paper. A related paper using state models and solutions to the Yang-Baxter equation is Oriented Quantum Algebras -- pdf Download, and a paper emphasizing the categorical algebra is Centrality and the KRH Invariant -- pdf Download.

Another two papers you should download are New Invariants -- pdf Download and Functional Integral and Vassiliev Invariants -- pdf Download . The first is an introduction to the bracket polynomial, Jones polynomial and related invariants. The second is an introduction to Vassiliev invariants, Witten's quantum field theory approach to link invariants and Kontsevich integrals.

Virtual knot theory is a generalization of classical knot theory that utilizes a crossing that "is not there." This virtual crossing can be interpreted as a detour through a handle that is attached to the plane of projection, and the theory of virtual knots can be understood as a theory of (stabilized) knots and links in thickend surfaces. Here are two papers on virtual knot theory: Virtual Knots - pdf download and Detecting Virtual Knots - pdf download .

See Bracket Calculations -- pdf Download for notes on skein calculations of bracket and Jones polynomial, and for a sketch of how to produce links that are not detectable by the Jones polynomial. This file is meant to be read in conjuntion with the next file on Mathematica calculations.

See Mathematica Calculates the Bracket Polynomial -- pdf Download for a Mathematica worksheet illustrating how to use a computer to calculate the bracket polynomial. The worksheet contains examples of calculations for the trefoil knot, the figure eight knot, the Hopf link, the knot 9_{42} (first example of a knot that is chiral but whose chirality cannot be detected by the Jones polynomial) and an example due to Morwen Thistlethwaite of a link of two components whose linking is invisible to the Jones polynomial. Read this worksheet in conjunction with the previous file on bracket calculations.

See From Tangle Fractions to DNA -- pdf Download for notes on rational tangles, rational knots and applications to DNA. This is a survey paper by LK and Sofia Lambropoulou. See Knots and DNA -- pdf Download for an essay on topology and DNA by DeWitt Sumners.