Upper bounds on the minimal length of cubic lattice knots

Abstract

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we consider every knot in the cubic lattice. Especially the minimal length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used "minimal edge number" instead) and proved that the minimal length of the trefoil knot $3_1$ is $24$. Also the minimal lengths of the knots $4_1$ and $5_1$ are known to be $30$ and $34$, respectively. In the article we find a general upper bound of the minimal length of a nontrivial knot $K$, except the trefoil knot, in terms of the minimal crossing number $c(K)$. The upper bound is $\frac{3}{2}c(K)^2 + 2c(K) + \frac{1}{2}$. Moreover if $K$ is a non-alternating prime knot, then the upper bound is $\frac{3}{2}c(K)^2 - 4c(K) + \frac{5}{2}$. Furthermore if $K$ is $(n+1,n)$-torus knot, then the upper bound is $6 c(K) + 2 \sqrt{c(K)+1} +6$.