
COMMENTS

A spatial polygon is a finite set of straight line segments in R3 which intersect only at their endpoints; the lines are called edges and their endpoints are called vertices; exactly two edges meet at every vertex. There must be at least 3 edges to make a triangle (the trivial knot) and it is not hard to show that a knotted polygon must have at least 6 edges. "Enumerating these polygons soon becomes impracticable because the number of cases explodes as n increases."
Hong et al. prove: "The lattice stick number s_L(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except trefoil knot, in terms of the minimal crossing number c(K) which is s_L(K) <= 3 c(K) + 2. Moreover if K is a nonalternating prime knot, then s_L(K) <= 3 c(K)  4". [Jonathan Vos Post, Sep 04 2012]


EXAMPLE

a(3) = 1 because the unique polygonal knot of 3 edges can be drawn with vertex coordinates (4,9,5), (7,9,5), (9,3,5).
a(6) = 1 because the unique polygonal knot of 6 edges can be drawn with vertex coordinates (4,9,5), (7,7,5), (7,9,5), (1,9,5), (9,3,5), (9,5,5).
a(7) = 1 because the unique polygonal knot of 7 edges can be drawn with vertex coordinates (9,6,3), (4,7,3), (1,7,2), (9,2,10), (4,5,10), (2,2,2), (5,2,5).
