For a nonzero vector v in R^{3}, let p_{v} denote perpendicular projection onto the span of v.
For a knot K, let b_{v} denote the number of components of the preimage of the set of local maximum values
of p_{v} restricted to K.

The bridge number of K is the minimum value of b_{v} taken over all nonzero v and all knots isotopic to K.

The super bridge index is the minimum over all knots isotopic to K of the maximum of b_{v}, taken over all v.

Randell observed that the super bridge index is bounded above by half the polygon index: superbridge(K) ≤ polygonnumber(K)/2 [4] (this is the source for the data in the table for knots with 9 or fewer crossings).

Upper bounds for the super bridge indices of knots follow from Randell's work [4], in which he shows that the super bridge index is bounded above by half the polygon index. This result supplies upper bounds for 8_{3, 5, 6, 7, 9, 10, 12, 15} and 9_{2-14, 16-39, 44, 47-49}.

Kuiper [3] determined the super bridge index of torus knots: 3_{1}, 5_{1},7_{1}, 8_{19}, 9_{1}, 10_{124}, and 11a_{367}.

9_{35},
9_{39},
9_{43},
9_{45},
9_{48},
11n_{71},
11n_{73},
11n_{74},
11n_{75},
11n_{76},
11n_{78},
11n_{81}.

Ref. [5]

[1] Jeon, C. B. and Jin, G. T., "A computation of superbridge index of knots" (English summary),
Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramifications **11** (2002), no. 3, 461-473.

[2] Jeon, C. B. and Jin, G. T., "There are only finitely many 3-superbridge knots," Knots in Hellas '98, Vol. 2 (Delphi).
J. Knot Theory Ramifications **10** no. 2 (2001), 331-343.

[3] Kuiper, N. H., "A new knot invariant," Math. Ann. **278** (1987), no. 1-4, 193-209.

[4] Randell, R., "Invariants of piecewise-linear knots," Knot theory (Warsaw, 1995), Banach Center Publ.,
**42**, Polish Acad. Sci., Warsaw (1998), 307-319.

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