Ropelengths of closed braids

Abstract

For a link K, let L ( K ) denote the ropelength of K and let Cr ( K ) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L ( K ) in terms of Cr ( K ) . It is well known that there exist positive constants c 1 , c 2 such that for any link K, c 1 ⋅ ( Cr ( K ) ) 3 / 4 ⩽ L ( K ) ⩽ c 2 ⋅ ( Cr ( K ) ) 3 / 2 . In this paper, we show 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 ) . Thus, if a link K admits a closed braid representation in which the number of crossings is bounded by a ( Cr ( K ) ) for some constant a ⩾ 1 , then we have L ( K ) ⩽ c ⋅ ( Cr ( K ) ) 6 / 5 for some constant c > 0 which only depends on a. In particular, this holds for any link that admits a reduced alternating closed braid representation, or any link K that admits a regular projection in which there are at most O ( Cr ( K ) ) crossings and O ( Cr ( K ) ) Seifert circles.