knot groups

PSL(2,C) Representations of Knot Groups

| chrisman.76

Kate Petersen (University of Minnesota Duluth)

I will discuss a method of producing defining equations for representation varieties of the canonical component of a knot group into PSL(2,C). This method uses only a knot diagram satisfying a mild restriction and is based upon the underlying geometry of the knot complement. In particular, it does not involve any polyhedral decomposition or triangulation of the link complement. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is joint work with Anastasiia Tsvietkova.

Knot groups and virtual knots

| chrisman.76

H. A. Dye (Mckendree U) and A. Kaestner (North Park U)

In the paper, Virtual parity Alexander polynomials, we defined a virtual knot group that used information about the parity of the classical crossings. This virtual knot group was defined using ad-hoc methods. In the paper, Virtual knot groups and almost classical knots, Boden et al describe several different knot groups obtained from virtual knots. These knot groups are related and specializations lead to the classical knot group. Here, we construct a formal structure for virtual knot groups and examine specializations and extensions of the groups.

References:

H. A. Dye and A. Kaestner, Virtual parity Alexander polynomials, https://arxiv.org/abs/1907.08709