In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are called ideal knots and ideal links respectively.

A numeric approximation of an ideal trefoil.

Definition

The ropelength of a knotted curve ${\displaystyle C}$ is defined as the ratio ${\displaystyle L(C)=\operatorname {Len} (C)/\tau (C)}$, where ${\displaystyle \operatorname {Len} (C)}$ is the length of ${\displaystyle C}$ and ${\displaystyle \tau (C)}$ is the knot thickness of ${\displaystyle C}$.

Ropelength can be turned into a knot invariant by defining the ropelength of a knot ${\displaystyle K}$ to be the minimum ropelength over all curves that realize ${\displaystyle K}$.

Ropelength minimizers

One of the earliest knot theory questions was posed in the following terms:

In terms of ropelength, this asks if there is a knot with ropelength ${\displaystyle 12}$. The answer is no: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least ${\displaystyle 15.66}$.^[1] However, the search for the answer has spurred research on both theoretical and computational ground. It has been shown that for each link type there is a ropelength minimizer although it may only be of differentiability class ${\displaystyle C^{1}}$.^[2][3] For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372.^[1]

Dependence on crossing number

An extensive search has been devoted to showing relations between ropelength and other knot invariants such as the crossing number of a knot. For every knot ${\displaystyle K}$, the ropelength of ${\displaystyle K}$ is at least proportional to ${\displaystyle \operatorname {Cr} (K)^{3/4}}$, where ${\displaystyle \operatorname {Cr} (K)}$ denotes the crossing number.^[4] There exist knots and links, namely the ${\displaystyle (k,k-1)}$ torus knots and ${\displaystyle k}$-Hopf links, for which this lower bound is tight. That is, for these knots (in big O notation),^[3]

$${\displaystyle L(K)=O(\operatorname {Cr} (K)^{3/4}).}$$

On the other hand, there also exist knots whose ropelength is larger, proportional to the crossing number itself rather than to a smaller power of it.^[5] This is nearly tight, as for every knot,

$${\displaystyle L(K)=O(\operatorname {Cr} (K)\log ^{5}(\operatorname {Cr} (K))).}$$

The proof of this near-linear upper bound uses a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice.^[6] However, no one has yet observed a knot family with super-linear dependence of length on crossing number and it is conjectured that the tight upper bound should be linear.^[7]

References