Upper bound on lattice stick number of knots

TL;DR

This paper establishes upper bounds on the lattice stick number of nontrivial knots, relating it to their minimal crossing number, with tighter bounds for non-alternating prime knots, advancing understanding of knot complexity in cubic lattices.

Contribution

It provides new upper bounds on the lattice stick number of knots based on crossing number, including improved bounds for non-alternating prime knots.

Findings

For nontrivial knots (except trefoil), s_L(K) ≤ 3 c(K) + 2.

For non-alternating prime knots, s_L(K) ≤ 3 c(K) - 4.

Bounds improve understanding of knot complexity in lattice models.

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 $s_L(K) \leq 3 c(K) - 4$.