For a link L, the braid index, denoted b(L), is the fewest number of strings needed to express L as a closed braid.
A theorem by Yamada shows that the braid index of L is equal to the minimum number of Seifert circles in a diagram of L .
The braid index is related to the bridge index, br(K), by br(K) ≤ b(K).
Stepan Orevkov determined the braid index and found minimum width braid representatives for most of the links in LinkInfo. Lower bounds for the braid index were found using the Morton-Franks-Williams Inequality based on the Jones polynomial. In a few cases, this inequality was applied to the link with its components doubled, replacing each component with two parallel components. Computations were done using a program written by Morton and Short. Upper bounds were found be building explicit braid representatives; most of these could be done using . The remaining cases were handled by using basic techniques to simplify given braid representatives.
 Jones, V. F. R., Hecke algebra representations for braid groups and link polynomials, Ann. of Math., 126 (1987), 335-388.
 Moran, S., The Mathematical Theory of Knots and Braids, An Introduction, Elsevier, New York (1983).
 Vogel, P., Representation of links by braids: a new algorithm, Comment. Math. Helv. 65 (1990), 104-113.
 Yamada, S., The Minimal Number of Seifert Circles Equals the Braid Index of a Link, Invent. Math. 89 (1987), 347-356.
 Diao, Yuanan; Hetyei, Gabor; Liu, Pengyu. The braid index of reduced alternating links, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 3, 415-434.