Every knot has a description as a polygon, determined by a finite sequence of vertices in three-space and formed by joining successive vertices by line segments. The stick number of a knot is the minimum number of vertices in such a polygonal description of the knot. Initial data for the table, through 9 crossings, came from [2], where the work of [1], [4], and [5] is credited for the original computations. The references include many values that do not appear in that original source.
935,
939,
943,
945,
948.
Ref. [3]
918,
1018,
1068,
1082,
1084,
1093,
10100,
10152.
Upper bounds of 10. Ref. [6]
1058,
1066,
1079,
1080.
Upper bounds of 11. Ref. [6]
11n72,
12n553.
Upper bounds of 11. Ref. [7]
11n77,
12n60,
12n219.
Upper bounds of 12. Ref. [7]
12n66,
12n225.
Upper bounds of 13. Ref. [7]
[1] Calvo, J. A., Geometric Knot Theory, Ph.D. Thesis, Univ. Calif. Santa Barbara, 1998.
(For polygons with up to 9 edges.)
[2] Cromwell, P., Knots and Links, Cambridge University Press, 2004.
[3] Eddy, T. D. and Shonkwiler, C., "New stick number Bounds from random sampling of confined polygons," Experimental Math., DOI: 10.1080/10586458.2021.1926000. Arxiv preprint.
[4] Negami, S., "Ramsey theorems for knots, links and spatial graphs," Trans. Amer. Math. Soc. 324 no. 2 (1991), 527-541.
[5] Randell, R., "Invariants of piecewise-linear knots," Knot theory (Warsaw, 1995), 307-319, Banach Center Publ.,
42, Polish Acad. Sci., Warsaw, 1998.
(For polygons with up to 8 edges.)
[6] Shonkwiler, C., "All prime knots through 10 crossings have superbridge Index ≤ 5," J. Knot Theory Ramifications, DOI: 10.1142/S0218216522500237. Arxiv preprint.
[7] Shonkwiler, C., "New superbridge index calculations from non-minimal realizations." Arxiv preprint.