Polygon Index

The polygon number of a knot is the minimum number of vertices in a polygonal description of the knot. Initial data for the table, through 9 crossings, came from Peter Cromwell's book, "Knots and Links," where the work of Randell, Negami, and Calvo is credited for the original computations.

Specific Knots

935, 939, 943, 945, 948.
   Ref. [4]

References

[1] Calvo, J. A., Geometric Knot Theory, Ph.D. Thesis, Univ. Calif. Santa Barbara, 1998.
(For polygons with up to 9 edges.)

[2] Negami, S., "Ramsey theorems for knots, links and spatial graphs," Trans. Amer. Math. Soc. 324 no. 2 (1991), 527-541.

[3] 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.)

[4] Eddy, T. D. and Shonkwiler, C., "New stick number Bounds from random sampling of confined polygons," Arxiv preprint.