3,6

A spatial polygon is a finite set of straght 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."

Peter Cromwell, Knots and Links, Cambridge University Press, 2004, Sec. 1.3 (pp. 5-8), Appendix E.

Robert G. Scharein, Stick numbers for minimal stick knots, Feb 15, 2004.

Bryson R. Payne, Advanced Knot Theory Topics, Knot Theory Online.

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).

Cf. A002863 Number of prime knots with n crossings.

Sequence in context: A035549 A137663 A161628 this_sequence A164917 A166238 A014197

Adjacent sequences: A122056 A122057 A122058 this_sequence A122060 A122061 A122062

hard,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Sep 14 2006