The Braid Length


For 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 that has 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 which yields the braid length is not necessarily the braid which 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 the briad length is not realized with a braid with the minimal number of strands, a minimizing braid in both senses is given in the braid description.

For 12 crossings 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

T. Gittings, Minimum Braids: A Complete Invariant of Knots and Links, Arxiv:math.GT/0401051

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