Ropelength
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. Definition. The ropelength of a knotted curve formula_1 is defined as the ratio formula_2, where formula_3 is the length of formula_1 and formula_5 is the knot thickness of formula_1. Ropelength can be turned into a knot invariant by defining the ropelength of a knot formula_7 to be the minimum  ropelength over all curves that realize formula_7. 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 formula_9. The answer is no: an argument using quadrisecants shows that the ropelength of any nontrivial knot has to be at least formula_10. 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 formula_11. For the simplest nontrivial  knot, the trefoil knot, computer simulations have shown that its minimum ropelength is at most 16.372. 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 formula_7, the ropelength of formula_7 is at least proportional to formula_14, where formula_15 denotes the crossing number. There exist knots and links, namely the formula_16 torus knots and formula_17-Hopf links, for which this lower bound is tight. That is, for these knots (in big O notation), formula_18 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. This is nearly tight, as for every knot, formula_19 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. 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.