Relating the untwisting and surgery description numbers

Geometry Topology Seminar
Monday, April 18, 2022 - 2:00pm for 1 hour (actually 50 minutes)
Samantha Allen – UGA
Jennifer Hom

The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. The surgery description number of K can be defined similarly, allowing for multiple full twists in a single twisting region. We can find no examples of knots in the literature where these two invariants are not equal. In this talk, I will provide the first known example where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  This will involve an exploration of the existing obstructions (often Heegaard-Floer theoretic) as well as the algebraic versions of these invariants.  In addition, we show the surprising result that the untwisting number of a knot is at most three times its surgery description number.  This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.