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 [5], where the work of [3], [11], and [13] is credited for the original computations. A substantial update, including stick number bounds for all knots through 13 crossings, came from [4].
When the exact stick number of a knot is known, a reference is given below for what seems to be the first source which found an example with that many segments.
31.
Ref. [11]
41.
Ref. [12]
51.
Ref. [8]
52,
62.
Ref. [Smith]
61,
63.
Ref. [10]
819.
Ref. [1]
10124.
Ref. [7]
71,
72,
73,
74,
75,
76,
77,
820,
821.
Ref. [9]
816,
817,
818,
929,
934,
940,
941,
942,
944,
946,
947,
949.
Ref. [14]
935,
939,
943,
945,
948.
Ref. [6]
13n592.
Ref. [2]
11n71,
11n75,
11n76,
11n78,
13n225,
13n230,
13n285,
13n288,
13n307,
13n584,
13n586,
13n593,
13n602,
13n603,
13n604,
13n607,
13n608,
13n1192,
13n5018.
Ref. [4]
[1] Adams, C., Brennan, B. M., Greilsheimer, D. L., and Woo, A. K., "Stick numbers and composition of knots and links," J. Knot Theory Ramifications 6 no. 2 (1997), 149–161.
[2] Blair, R., Eddy, T. D., Morrison, N., and Shonkwiler, C. "Knots with exactly 10 sticks," J. Knot Theory Ramifications 29 no. 3 (2020), 2050011.
[3] Calvo, J. A., Geometric Knot Theory, Ph.D. Thesis, Univ. Calif. Santa Barbara, 1998.
[4] Cantarella, J., Rechnitzer, A., Schumacher, H., and Shonkwiler, C., "New upper bounds for stick numbers." Arxiv preprint.
[5] Cromwell, P., Knots and Links, Cambridge University Press, 2004.
[6] Eddy, T. D. and Shonkwiler, C., "New stick number Bounds from random sampling of confined polygons," Experimental Math. 31 no. 4 (2021), 1373–1395.
[7] Jin, G. T., "Polygon indices and superbridge indices of torus knots and links," J. Knot Theory Ramifications 6 no. 2 (1997), 281–289.
[8] Jin, G. T. and Kim, H.S., "Polygonal knots," J. Korean Math. Soc. 30 no. 2 (1993), 371–383.
[9] Meissen, M., "Edge number results for piecewise-linear knots," Knot theory (Warsaw, 1995), 235–242, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998.
[10] Millett, K. C., "Knotting of regular polygons in 3-space," J. Knot Theory Ramifications 3 no. 3 (1994), 263–278.
[11] Negami, S., "Ramsey theorems for knots, links and spatial graphs," Trans. Amer. Math. Soc. 324 no. 2 (1991), 527-541.
[12] Randell, R., "An elementary invariant of knots," J. Knot Theory Ramifications 3 no. 3 (1994), 279–286.
[13] Randell, R., "Invariants of piecewise-linear knots," Knot theory (Warsaw, 1995), 307-319, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998.
[14] Scharein, R.G., Interactive Topological Drawing, Ph.D. thesis, Univ. British Columbia, 1998.
[15] Smith, K.F., Generalized Braid Arrangements and Related Quotient Spaces, Ph.D. thesis, Univ. Iowa, 1992.