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In-Person and Online TalkÌý

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Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focusedÌýon the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned withÌýinvariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology.ÌýIn my talk, I will describe a new relationship between these two fields that was discovered with the aidÌýof machine learning. Specifically, we show that the knot signature can be estimated surprisingly accuratelyÌýin terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of aÌýhyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is thatÌýtwice the knot signature and the natural slope differ by at most a constant times the hyperbolic volumeÌýdivided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus.ÌýWe will also present a refined version of the inequality where the upper bound is a linear function of the volume,Ìýand the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot.ÌýMy talk will outline the proofs of these results, as well as describing the role that machine learning playedÌýin their discovery.