Braid Length

For a knot K, the braid length is the fewest number of crossings needed to express K as a closed braid.

Let D be a diagram for a knot K with n crossings and m Seifert circles. The braid length of K has the following upper bound: braid length (K) ≤ n + (m-1)(m-2).

The closed braid that yields the braid length is not necessarily the braid that yields the braid index. The first example of such a knot is 10136. This knot has braid length 10, which is achieved in a 5-strand braid. However, the knot has braid index 4, since the knot is isotopic to a braid with 4 strands and 11 crossings. In cases in which a braid with the minimal number of strands does not give the braid length, a minimizing braid in both senses is given in the braid description.

For 12 crossing knots the initial data was supplied by Stoimenow. Gittings found several corrections and identified the cases in which the minimum braid length does not occur with the minimally stranded braid.

References

[1] Gittings, T., Minimum Braids: A Complete Invariant of Knots and Links, Arxiv preprint.

[2] Vogel, P., Representation of links by braids: a new algorithm, Comment. Math. Helv. 65 (1990), 104-113.