Link lengths and their growth powers

Abstract

For a certain infinite family f of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every K is an element of f. It is known that the stick number and lattice stick number grow between the 1/2 and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the 3/4 power of crossing number (which is called the four-thirds power law). Furthermore, the minimal lattice length and minimum ropelength grow at most as O(c(K)[ln(c(K))](5)), but it is unknown whether any family exhibits superlinear growth. For any real number r between 1/2 and 1, we give an infinite family of non-splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number. Furthermore for any real number r between 3/4 and 1, we give another infinite family of non-splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number.