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 [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.

Specific Knots

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]

References

[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.

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