Extra Seminar - Dr. R. van der Veen

TITLE: Geometry of knots and quantum groups

ABSTRACT

The purpose of this talk is to show how a knotted circle K in 3-space X gives rise to a wealth of geometrical objects and questions. Usually the complement of the knot X - K carries a unique metric with constant curvature -1. Through holonomy it follows that the fundamental group G of the complement may be represented faithfully into SL(2,C). In other words the variety of representations of G into SL(2) contains a unique distinguished point up to conjugation.

In principle all of the above may still be carried out if one replaces the Lie group SL(2) by a quantum group. By a quantum group we mean a non-commutative Hopf algebra deformation of the coordinate ring of SL(2). This is natural to do since the operations available on knots are precisely those found in quantum groups. Recently the technique of solvable approximation allowed us to make effective (polynomial time) computations in this setting, opening up a whole new field of research. Potentially this will resolve the long standing slice-ribbon conjecture on locally flat disks in four-manifolds.

The slides and handout for this talk will be available at http://www.rolandvdv.nl/Talks/Groningen18.html