Braid Index

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 [4].

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 [5]. The remaining cases were handled by using basic techniques to simplify given braid representatives.

References

[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.

[5] Diao, Yuanan; Hetyei, Gabor; Liu, Pengyu. The braid index of reduced alternating links, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 3, 415-434.