On the complexity of braids

Details On the complexity of braids On the complexity of braids

2004-03-10

arxiv.org/abs/math/0403177v2

Mathematics

Geometric Topology

Version 2: added section on Teichmueller geometry, removed section on train tracks

Scientific paper

We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $\Delta\_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $\Delta\_{ij}^k$ as $\log(|k|+1)$ instead of simply $|k|$. The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. This gives rise to a new combinatorial model for the Teichmueller space of an $n+1$ times punctured sphere. We also show how to recover a braid from its curve diagram in polynomial time. The key r\^ole in the proofs is played by a technique introduced by Agol, Hass, and Thurston.

Affiliated with

Also associated with

No associations

Rating

On the complexity of braids does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the complexity of braids, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the complexity of braids will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-418682