Knots with small lattice stick numbers

TL;DR

This paper proves that only the trefoil and figure-8 knots have lattice stick numbers less than 15, confirming previous numerical estimates through rigorous mathematical proof.

Contribution

It provides a rigorous mathematical proof identifying the only knot types with lattice stick numbers under 15, specifically the trefoil and figure-8 knots.

Findings

Trefoil knot and figure-8 knot are the only knots with lattice stick number less than 15.

Validated previous numerical estimations with a formal proof.

Established a boundary for lattice stick numbers of knots.

Abstract

The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot $3_1$ and the figure-8 knot $4_1$ are the only knot types of lattice stick number less than 15, which verifies the result from previous numerical estimations on this quantity.