A theorem of Yamada shows that the braid index of K is equal to the minimum number of Seifert circles in a diagram of K.
The braid index is related to the bridge index, br(K), by the following inequality: br(K) ≤ b(K).
The values of the braid index are taken from the reference by Jones for knots with fewer then 11 crossings, and for 11-and 12-crossing knots, the data was provided to us by Stoimenow.
V. F. R. Jones, Hecke algebra representations for braid groups and link polynomials, Ann. of Math., 126 (1987) 335-388.
S. Moran, The Mathematical Theory of Knots and Braids, An Introduction, Elsevier, New York: 1983.
P. Vogel, Representation of links by braids: a new algorithm, Comment. Math. Helv. 65 104-113, 1990.
S. Yamada, The Minimal Number of Seifert Circles Equals the Braid Index of a Link, Invent. Math. 89 347-356, 1987.