Month: March 2022

PSL(2,C) Representations of Knot Groups

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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.

Geometry of alternating links on surfaces

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Jessica Purcell (Monash University)

It is typically hard to relate the geometry of a knot complement to a diagram of the knot, but over many years mathematicians have been able to relate geometric properties of classical alternating knots to their diagrams. Recently, we have modified these techniques to investigate geometry of a much wider class of knots, namely alternating knots with diagrams on general surfaces embedded in general 3-manifolds. This has resulted in lower bounds on volumes, information on the geometry of checkerboard surfaces, restrictions on exceptional Dehn fillings, and other geometric properties. However, we were unable to extend upper volume bounds broadly. In fact, recently we showed an upper bound must depend on the 3-manifold in which the knot is embedded: We find upper bounds for virtual knots, but not for other families. We will discuss this work, and some remaining open questions. This is joint in part with Josh Howie and in part with Effie Kalfagianni.