[Geometry, Topology and Dynamics Seminar] Alternating knots are rare by Dr. Harrison Chapman (Colorado State University)
Monday, February 18, 2019 - 16:00 to 17:00
Lab 3, B700
Title: Alternating knots are rare
Abstract: It has long been suspected that only a very small subset of knots are
alternating. Thistlethwaite's remarkable 1998 proof that alternating links are
exponentially rare yields strong evidence for the rarity of alternating knots,
but does not prove it. Thistlethwaite's argument relies on two remarkable
facts: First that alternating links satisfy Tait's flyping conjecture (work
with Menasco), and second that alternating links can be enumerated
(work with Sundberg). While the flyping conjecture still applies to
alternating knots, no enumeration of alternating knots yet exists. Using a
pattern theorem for alternating knots, we can avoid the need for precise
enumeration and conclusively show that alternating knots are in fact rare.
We'll discuss both the proof of the rarity of alternating links and how to
extend this into a proof of the rarity of alternating knots.