The ropelength of knots

Abstract: The ropelength $R(K)$ of a knot $K$ is the minimum length of a unit thickness rope needed to tie the knot. If $K$ is alternating, it is conjectured that $R(K)\ge a {\rm{Cr}}(K)$ for some constant $a>0$, where ${\rm{Cr}}(K)$ is the minimum crossing number of $K$. In this talk I will first give a brief introduction to the ropelength problem. I will then show that there exists a constant $a_0>0$ such that $R(K)\ge a_0 \textbf{b}(K)$ for any knot $K$, where $\textbf{b}(K)$ is the braid index of $K$. It follows that if $\textbf{b}(K)\ge a_1 {\rm{Cr}}(K)$ for some constant $a_1>0$, then $R(K)\ge a_0 a_1 {\rm{Cr}}(K)=a {\rm{Cr}}(K)$. However if $\textbf{b}(K)$ is small compared to ${\rm{Cr}}(K)$ (in fact there are alternating knots with arbitrarily large crossing numbers but fixed braid indices), then this result cannot be applied directly. I will show that this result can in fact be applied in an indirect way to prove that the conjecture holds for a large class of alternating knots, regardless what their braid indices are.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Topology and geometry: extremal and typical

Series comments: Sign up for the mailing list groups.google.com/g/tget-seminar to receive zoom links.