Hyperbolicity and Turaev hyperbolicity of virtual knots

Colin Adams (Williams College)

Abstract: Hyperbolic volume has been a powerful invariant for classical knots. In this talk we extend it to virtual knots, including calculations of virtual knot volumes. We further introduce Turaev volume for knots and virtual knots and show that EVERY knot has a Turaev volume.

References:

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Turaev Hyperbolicity of Classical and Virtual Knots, https://arxiv.org/pdf/1912.09435.pdf

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Tg-Hyperbolicity of Virtual Links, https://arxiv.org/pdf/1904.06385.pdf

3 thoughts on “Hyperbolicity and Turaev hyperbolicity of virtual knots

  1. Thank you Prof. Adams for an interesting talk! To ask our speaker a question, reply to this post.

    1. You mention in your talk that one must be careful about what is meant by “prime virtual knot”, but do not elaborate further. There is a well-known definition of “prime virtual knot” (due to Matveev) and it is easy to construct examples of hyperbolic knots in thickened surfaces which are not prime in this sense. Could you elaborate on the difference between your definition of prime and Matveev’s definition?

      1. Matveev defines a virtual knot to be prime if there is no vertical annulus in the thickened surface that intersects the knot twice such that the knot is not parallel into the annulus to either side . As you mention, such an annulus is not an obstruction to hyperbolicity, since it appears in the knot complement as a twice-punctured annulus, and we only need worry about unpunctured annuli as obstructions.

        Our definition of prime is that there is no ball in the thickened surface with boundary intersecting the knot twice such that to the inside, the arc of the knot is nontrivial. This is an obstruction to hyperbolicity since its boundary in the knot complement is an essential annulus.

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