A122059 Number of different polygonal knots with n straight line segments.

1, 0, 0, 1, 1, 2, 3, 0, 4

Offset: 3

Author

Jonathan Vos Post, Sep 14 2006

Keywords

• hard
• more
• nonn

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 non-alternating prime knot, then s_L(K) <= 3 c(K) - 4". - Jonathan Vos Post, Sep 04 2012

Examples

			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).
		

References

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

Links

• KyungPyo Hong, SungJong No, SeungSang Oh, Upper bound on lattice stick number of knots, arXiv:1209.0048v1 [math.GT], Sep 01 2012
• Bryson R. Payne, Advanced Knot Theory Topics, Knot Theory Online
• Robert G. Scharein, Stick numbers for minimal stick knots, Feb 15 2004

Crossrefs

Cf. A002863 (number of prime knots with n crossings).