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

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 ($4_1$).