Ropelengths of closed braids
The paper deals with geometric knot theory, particularly with the relationships between the ropelength \(L(K)\) of a link \(K\) (how long a unit thickness rope of \(K\) needs to be) and its crossing number \(Cr(K)\). It is known that, for any knot \(K\), a lower bound for \(L(K)\) is given by \(O((Cr(K))^{3/4})\) and that this bound is achievable for some prime knot families. On the contrary, no sharp upper bound for \(L(K)\) in terms of \(Cr(K)\) has been found in general; the best upper bound for the ropelength of any knot \(K\) is of the order \(O((Cr(K))^{3/2})\). In the present paper, the authors prove that any closed braid with \(n\) crossings can be realized by a unit thickness rope of length at most of the order \(O(n^{6/5})\); this theorem still does not improve the \(3/2\) power bound in general, since the number of crossings in a (minimum) braid representation of a knot may be larger than its crossing number. However, for all knots \(K\) admitting a braid representation in which the number of crossings is of the order \(O(Cr(K))\) (in particular, if the closed braid is in a reduced alternating form), the present result provides a much better upper bound on their ropelength, in terms of their crossing number \(Cr(K)\).
scientific article; zbMATH DE number 5123634