Geometry of alternating links on surfaces

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.

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