Super Bridge Index

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.

The bridge number of K is the minimum value of bv 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 bv, taken over all v.

Randell observed that the super bridge index is bounded above by half the polygon index: superbridge(K) ≤ polygonnumber(K)/2 [5] (this is the source for the data in the table for knots with 9 or fewer crossings).

Specific Knots

Upper bounds for the super bridge indices of knots follow from Randell's work [5], 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 [4] determined the super bridge index of torus knots: 31, 51,71, 819, 91, 10124, and 11a367.

Shonkwiler computed the values for 41 knots of 9 and fewer crossings and 14 higher-crossing knots [1, 7, 8] and found bounds for 10-crossing knots [6].

9{35, 39, 43, 45, 48}, 11n{71, 73, 74, 75, 76, 78, 81}.
   Ref. [1]

8{1–3, 5–8, 10–15}, 9{7, 16, 20, 26, 28, 32, 33}.
   Ref. [7]

9{3, 4, 6, 9, 11, 13, 17, 18, 22, 23, 25, 27, 30, 31, 36}, 11n{72, 77}, 12n{60, 66, 219, 225, 553}.
   Ref. [8]

1058, 1066, 1080.
    Have superbridge index ≤ 5. Ref. [6]

References

[1] Eddy, T. D. and Shonkwiler, C., "New stick number bounds from random sampling of confined polygons," Experimental Math., DOI: 10.1080/10586458.2021.1926000. Arxiv preprint.

[2] 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.

[3] 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.

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

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

[6] Shonkwiler, C., "All prime knots through 10 crossings have superbridge Index ≤ 5," J. Knot Theory Ramifications, DOI: 10.1142/S0218216522500237. Arxiv preprint.

[7] Shonkwiler, C., "New computations of the superbridge index," J. Knot Theory Ramifications 29 (2020), no. 14, 2050096. Arxiv preprint.

[8] Shonkwiler, C., "New superbridge index calculations from non-minimal realizations." Arxiv preprint.