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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Knot theory and machine learning - Marc Lackenby\,
Oxford
DTSTART;TZID=Europe/London:20220316T160000
DTEND;TZID=Europe/London:20220316T170000
UID:TALK168167AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/168167
DESCRIPTION:Knot theory is divided into several subfields. One
of these is hyperbolic knot theory\, which is foc
used on the hyperbolic structure that exists on ma
ny knot complements. Another branch of knot theory
is concerned with invariants that have connection
s to 4-manifolds\, for example the knot signature
and Heegaard Floer homology. In my talk\, I will d
escribe a new relationship between these two field
s that was discovered with the aid of machine lear
ning. Specifically\, we show that the knot signatu
re can be estimated surprisingly accurately in ter
ms of hyperbolic invariants. We introduce a new re
al-valued invariant called the natural slope of a
hyperbolic knot in the 3-sphere\, which is defined
in terms of its cusp geometry. Our main result is
that twice the knot signature and the natural slo
pe differ by at most a constant times the hyperbol
ic volume divided by the cube of the injectivity r
adius. This theorem has applications to Dehn surge
ry and to 4-ball genus. We will also present a ref
ined version of the inequality where the upper bou
nd is a linear function of the volume\, and the sl
ope is corrected by terms corresponding to short g
eodesics that have odd linking number with the kno
t. My talk will outline the proofs of these result
s\, as well as describing the role that machine le
arning played in their discovery.\n\nThis is joint
work with Alex Davies\, Andras Juhasz\, and Nenad
Tomasev.
LOCATION:MR13
CONTACT:Henry Wilton
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