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SUMMARY:Knot theory and machine learning - Marc Lackenby\, Oxford
DTSTART:20220316T160000Z
DTEND:20220316T170000Z
UID:TALK168167@talks.cam.ac.uk
CONTACT:Henry Wilton
DESCRIPTION:Knot theory is divided into several subfields. One of these is
  hyperbolic knot theory\, which is focused on the hyperbolic structure tha
 t exists on many knot complements. Another branch of knot theory is concer
 ned with invariants that have connections to 4-manifolds\, for example the
  knot signature and Heegaard Floer homology. In my talk\, I will describe 
 a new relationship between these two fields that was discovered with the a
 id of machine learning. Specifically\, we show that the knot signature can
  be estimated surprisingly accurately in terms of hyperbolic invariants. W
 e introduce a new real-valued invariant called the natural slope of a hype
 rbolic knot in the 3-sphere\, which is defined in terms of its cusp geomet
 ry. Our main result is that twice the knot signature and the natural slope
  differ by at most a constant times the hyperbolic volume divided by the c
 ube of the injectivity radius. This theorem has applications to Dehn surge
 ry and to 4-ball genus. We will also present a refined version of the ineq
 uality where the upper bound is a linear function of the volume\, and the 
 slope is corrected by terms corresponding to short geodesics that have odd
  linking number with the knot. My talk will outline the proofs of these re
 sults\, as well as describing the role that machine learning played in the
 ir discovery.\n\nThis is joint work with Alex Davies\, Andras Juhasz\, and
  Nenad Tomasev.
LOCATION:MR13
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