(Department of Mathematics, McMaster University)
Virtual Knots and Their Alexander Invariants
|Date||Friday, October 31, 2014|
In this talk we will give a brief introduction to virtual knot theory: an up-and-coming field with roots in geometric topology, knot theory, and combinatorics. As in the classical case, algebraic invariants of virtual knots such as the knot group and the augmented knot group carry intrinsic topological information about the knot. We can use Alexander invariants to extract that information from the group structure. In virtual knot theory two augmented knot groups arise naturally, we will show that they are isomorphic.
Virtual knot theory is a rich field and has many specializations. Among virtual knots, we can study those that admit virtual knot diagrams that have Alexander numberings. These knots form a special class of virtual knots- known as almost classical knots. Almost classical knots share many similarities with classical knots. For almost classical knots, the augmented knot group is determined by the classical knot group and by work of Nakamura-et-al., its first elementary ideal is principal. This leads us to define the Conway potential function, a normalized version of the Alexander polynomial, for almost classical knots.
We will conclude with a brief discussion of the state of virtual knot theory and indicate possible future lines of research.