April | 2020 | classical knots, virtual knots, and algebraic structures related to knots

Hyperbolicity and Turaev hyperbolicity of virtual knots

April 27, 2020April 24, 2020 | chrisman.76

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

Dehn surgery on links vs the Thurston norm

April 20, 2020April 17, 2020 | chrisman.76

Maggie Miller (Princeton University)

Abstract: David Gabai showed that a minimum-genus surface in a knot complement remains minimum-genus when capped off into the zero-surgery on that knot, implying that surgery on a nontrivial knot can never yield $S^1\times S^2$ (thus proving the Property R conjecture). I study this problem for links. In particular, I show that if $L$ is a 2-component link with nonzero linking number and nondegenerate Thurston norm on its complement, then there exists a finite set $E\subset H_2(S^3\setminus\nu(L),\partial;\mathbb{R})$ so that if $S$ is norm-minimizing and not in $E$ up to scalar multiplication, then $\hat{S}$ is norm-minimizing in the 3-manifold obtained from $S^3$ by doing Dehn surgery on $L$ according to $\partial S$ . (The result generally holds for $n>1$ -component links with $E$ $(n-2)$ -dimensional.) The proof involves constructing a taut foliation on $S^3\setminus\nu(L)$ with nice boundary properties, motivated by Gabai’s proof of the Property R conjecture.