For a nonzero vector v in R3, let pv denote perpendicular projection onto the span of v. For a knot K, let bv denote the number of components of the preimage of the set of local maximum values of pv restricted to K. Randell, referenced below, observed that the super bridge index is bounded above by twice the polygon index.
The bridge number of K is the minimum value of bv taken over all nonzero v and all knots isotopic to K.
The superbridge index is the minimum over all knots isotopic to K of the maximum of bv, taken over all v.
Randell made the observation that superbridge(K) ≤ polygonnumber(K)/ 2
C. B. Jeon and G. T. Jin, There are only finitely many 3-superbridge knots, Knots in Hellas '98, Vol. 2 (Delphi). J. Knot Theory Ramifications 10 (2001), no. 2, 331--343.
C. B. Jeon and G. T. Jin, A computation of superbridge index of knots, (English. English summary) Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramifications 11 (2002), no. 3, 461--473.
N. H. Kuiper, A new knot invariant, Math. Ann. 278 (1987), no. 1-4, 193--209.
R. Randell, Invariants of piecewise-linear knots, Knot theory (Warsaw, 1995), 307--319, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998. (This is the source for the data in the table for knots with 9 or fewer crossings.)