Minimum lattice length and ropelength of knots

Abstract

Let Len(K) be the minimum length of a knot on the cubic lattice (namely the minimumlength necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len(K) of a nontrivial knot K in terms of its crossing number c(K) as follows: Len(K) <= min {3/4c(K)(2) + 5c(K) + 17/4, 5/8c(K)(2) + 15/2c(K) + 71/8}. The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the minimum ropelength Rop(K) which is close to twice Len(K): Rop(K) <= min{1.5c(K)(2) + 9.15c(K) + 6.79, 1.25c(K)(2) + 14.58c(K) + 16.90}.