Invited Speaker
Claus Ernst
The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers: Part 2
Abstract: For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. Here we show that there exists a constant a > 0 such that L(K) ≤ a Cr(K) ln5 (Cr(K)) for any K, that is, the ropelength upper bound of any knot is almost linear in terms of its minimum crossing number and is a significant improvement over the best known upper bound established previously, where it was shown that L(K) ≤ O((Cr(K)(3/2))).
In this part, we outline the reconstruction process of a plane graph (namely our original minimum knot projection) from the pieces of sub plane graphs obtained by subdividing the original graph repeatedly. An analysis of the volume the reconstructed graph occupies will yield the desired ropelength bound.
Authors: Yuanan Diao, Claus Ernst, Attila Por and Uta Ziegler
Presenter: Claus Ernst