The ropelengths of knots are almost linear in terms of their crossing numbers

Abstract

For a knot or link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text] and [Formula: see text] be the crossing number of [Formula: see text]. In this paper, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form [Formula: see text]. The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of [Formula: see text] is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order [Formula: see text]. The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).