Stick Number

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.

Specific Knots

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.