Invited Speaker
Yuanan Diao
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 $L(\K)\le a Cr(\K) \ln^5 (Cr(\K))$ 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 $L(\K)\le O((Cr(\K)^{\frac{3}{2}}))$. \medskip In this part, we lay out some basic graph theoretical results on subdividing a plane graph into sub plane graphs which are needed for constructing lattice knots of the given knot type with a length at most of order $O(n\ln ^5(n))$ where $n$ is the minimum crossing number of the given knot.