© | << < ? > >> | Dror Bar-Natan: Talks:

Yarn-Ball Knots

[K-OS], October 21, 2021

With Itai Bar-Natan, Iva Halacheva, and Nancy Scherich

Agenda. A modest light conversation on how knots should be measured.

Abstract. Let there be scones! Our view of knot theory is biased in favour of pancakes.

Technically, if $K$ is a 3D knot that fits in volume $V$ (assuming fixed-width yarn), then its projection to 2D will have about $V^{4/3}$ crossings. You'd expect genuinely 3D quantities associated with $K$ to be computable straight from a 3D presentation of $K$. Yet we can hardly ever circumvent this $V^{4/3}\gg V$ "projection fee". Exceptions include linking numbers (as we shall prove), the hyperbolic volume, and likely finite type invariants (as we shall discuss in detail). But knot polynomials and knot homologies seem to always pay the fee. Can we exempt them?

Warning. I made a silly mistake towards the end of the talk, which changes some of the numbers throughout the talk but does not change anything qualitative (namely, for finite type, 3D seems to beat 2D, with the same algorithms). A correction will be posted soon.

Handout: YarnBallKnots.pdf

Annotated Slides: YarnBallKnots@.pdf