The aim of the seminar is to understand various invariants of knots, links and tangles. Starting with an elementary aproach and polynomial invariants (Jones polynomial and variants), we should step by step learn more about quantum groups and R-matrices, Khovanov homology and categorification, and hopefully also about connections to statistical physics, cobordisms and Witten's work.
Time and place: Friday 10.40am-12.10am in the seminar room of the Mathematical Institute.
Program for March 14, 2014: Khovanov homology continued (M. Doubek).
The preliminary plan for the seminar for the coming semester or two is as follows:
- Jones polynomial defined using the skein relation (elementary approach).
- Quantum groups and R-matrices.
- R-matrices in statistical physics
- Invariants of tangles using ribbon categories, Jones polynomial as a special case.
- Other invariants obtained in that way: colored Jones polynomial, HOMFLY polynomial.
- Khovanov homology - an elementary approach, relation to Jones polynomial.
- Khovanov homology as a product of categorification of linear invariants of tangles.
- Kauffman R-matrices and their significance in physics.
- Witten's work.
- M. Doubek (Oct 18, 2013): Knots, links and isotopy, diagrams and isotopy of diagrams, Reidemeister moves, Kauffman bracket, Jones polynomial, framed knots and links.
- M. Doubek (Oct 25, 2013): The category TanC of C-labelled tangles and the ribbon category structure on it.
- M. Doubek (Nov 1, 2013): Schum's theorem about the freeness of TanC and a general method for constructing invariants of framed oriented tangles and links.
- R. O'Buachalla (Nov 8, 2013): Quasi-triangular bialgebras and the ribbon category structure for the modules over them.
- R. O'Buachalla (Nov 15, 2013): Co-quasi-triangular structures on bialgebras and the corresponding braided monoidal structures on comodules over them. Dual pairings of bialgebras and the relation to the (co)quasi-triangular structures. Constructing co-quasi-triangular bialgebras from solutions of the quantum Yang-Baxter equation.
- R. O'Buachalla (Nov 22, 2013): Ribbon bialgebras and Hopf algebras, the category of finite dimensional modules as a ribbon category. Uq(sl2) as an example.
- M. Doubek and R. O'Buachalla (Nov 29, 2013): Recovering the Jones polynomial from Schum's theorem and representation theory of Uq(sl2).
- L. Křižka (Dec 6, 2013): Knizhnik-Zamolodchikov equations for a complex semisimple Lie group g and the Drinfeld ribbon category D(g,κ).
- B. Jurčo (Dec 13, 2013): Knot invariants in statistical mechanical models: the vertex models (after V. F. R. Jones, see also Chari & Pressley and Kauffman in the literature).
- B. Jurčo (Jan 10, 2014): Knot invariants in statistical mechanical models, continued.
- M. Doubek (Feb 21, 2014): Khovanov homology - definition and computation for Hopf link (see the literature for on-line sources).
- M. Doubek (Feb 28, 2014): Khovanov homology continued. Proof that the differential squares to zero via the link to 2D topological quantum field theories.
The basic theory on linear invariants of tangles and the role of quantum groups is taken from:
- Ch. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, Panoramas et Synthèses, 5. Soc. Math. de France, Paris, 1997.
- A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
Other sources for quantum groups and their representation theory include:
- Ch. Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer-Verlag, New York, 1995.
- J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, AMS, Providence, RI, 1996.
The way from ordinary universal enveloping algebras U(g) to their quantized versions via the solutions of Knizhnik-Zamolodchikov equations is explained here:
- B. Bakalov, A. A. Kirillov, Jr., Lectures on tensor categories and modular functors, University Lecture Series 21, AMS, Providence, RI, 2001.
- P. I. Etingof, I. B. Frenkel, A. A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs 58, AMS, Providence, RI, 1998.
The connection to statistical mechanics was established by V. F. R. Jones in the paper
- V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), no. 2, 311-334.
and is explained also in the following two monographs:
- V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
- L. H. Kauffman, Knots and physics, fourth edition, Series on Knots and Everything 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
An introduction to Khovanov homology was taken from the following sources which are available on-line: