For a knot K, the braid index, denoted b(K), is the fewest number of strings needed to express K as a closed braid.
A theorem by Yamada shows that the braid index of K is equal to the minimum number of Seifert circles in a diagram of K [4].
The braid index is related to the bridge index, br(K), by br(K) ≤ b(K).
The values of the braid index in our table are taken from [1] for knots with fewer than 11 crossings, and for 11-and 12-crossing knots, the data was provided to us by Stoimenow.
[1] Jones, V. F. R., Hecke algebra representations for braid groups and link polynomials, Ann. of Math., 126 (1987), 335-388.
[2] Moran, S., The Mathematical Theory of Knots and Braids, An Introduction, Elsevier, New York (1983).
[3] Vogel, P., Representation of links by braids: a new algorithm, Comment. Math. Helv. 65 (1990), 104-113.
[4] Yamada, S., The Minimal Number of Seifert Circles Equals the Braid Index of a Link, Invent. Math. 89 (1987), 347-356.