Upper bound on lattice stick number of knots

Abstract

The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot $K$, except trefoil knot, in terms of the minimal crossing number $c(K)$ which is $s_L(K) \leq 3 c(K) +2$. Moreover if $K$ is a non-alternating prime knot, then $s_L(K) \leq 3 c(K) - 4$.