Bounds in simple hexagonal lattice and classification of 11-stick knots

TL;DR

This paper establishes bounds for the minimal stick and edge numbers of knots in the simple hexagonal lattice, showing they are less than in the cubic lattice, and classifies all 11-stick knots in this lattice.

Contribution

It introduces a linear transformation between lattices to compare knot complexities and classifies all 11-stick knots in the sh-lattice.

Findings

Stick and edge numbers in sh-lattice are less than in cubic lattice.

Only trefoil and figure-eight knots have 11 sticks in sh-lattice.

Abstract

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$).