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%I #20 Feb 08 2022 08:08:24
%S 1,0,0,1,1,2,3,0,4
%N Number of different polygonal knots with n straight line segments.
%C 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."
%C 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
%D Peter Cromwell, Knots and Links, Cambridge University Press, 2004, Sec. 1.3 (pp. 5-8), Appendix E.
%H KyungPyo Hong, SungJong No, SeungSang Oh, <a href="http://arxiv.org/abs/1209.0048">Upper bound on lattice stick number of knots</a>, arXiv:1209.0048v1 [math.GT], Sep 01 2012
%H Bryson R. Payne, <a href="http://www.oglethorpe.edu/faculty/~j_nardo/knots/advanced.htm">Advanced Knot Theory Topics</a>, Knot Theory Online
%H Robert G. Scharein, <a href="http://www.colab.sfu.ca/KnotPlot/sticknumbers/">Stick numbers for minimal stick knots</a>, Feb 15 2004
%e 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).
%e 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).
%e 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).
%Y Cf. A002863 (number of prime knots with n crossings).
%K hard,more,nonn
%O 3,6
%A _Jonathan Vos Post_, Sep 14 2006
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