Knots with small lattice stick numbers

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 31 and in figure 8 knot 41 are the only knot types of lattice stick number less than 15, which verifies the result from previous numerical estimations on this quantity.