Sunday Jan 20 14.00 Math 110: "Knots and Functional Integration without integration"
Tuesday Jan 22 14.00 Math 209: "Hopf algebras and Invariants of Links and Three-Manifolds"
(University of Illinois at Chicago)
Virtual Knot Theory
Abstract: Virtual knot theory is a generalization of classical knot theory. It bears the same relationship to classical knots as graphs bear to planar graphs. In representing a non-planar graph as a drawing in the plane we need to introduce crossings of certain edges that are not part of the graphical structure. These extra or virtual crossings are "not really there." Just so, virtual knot and link diagrams have crossings that are not really there. The virtual knots are represented by diagrams and they stand for the oriented Gauss codes (or Gauss diagrams) taken up to the equivalence relation generated by the Reidemeister moves.
In studying virtual knots one gets to look at knot theory in a new light. There are non-trivial virtual knots with unit Jones polynomial and non-trivial fundamental group. There are virtual knots with integer fundamental group and non-trivial Jones polynomial. There are virtual knots whose knottedness is so far undetected.
Virtual knots can be interpreted as knots in thickened surfaces taken up to the addition and subtraction of empty handles. This interpretation provides a direct connection of the virtual theory with the realities of three-dimensional topology.
This talk will introduce virtual knots and links and discuss the construction of invariants, topological interpretations, examples and open problems.