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# Yarn-Ball Knots

### The Fields Institute, November 13-15, 2020.

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 probably include the hyperbolic volume and certainly include finite type invariants (as we shall prove). But knot polynomials and knot homologies seem to always pay the fee.

Handout: YarnBallKnots.pdf

Annotated Slides: YarnBallKnots@.pdf