On the complexity of braids
By I. Dynnikov, Bert Wiest. 2004
70 citations recorded by the source corpus.
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 \Tei 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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8b16cb92e8b1bb2846ece23f70448b50c647781f
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Core Challenges of Social Robot Navigation: A Survey (2021, methodology): “[168, 170] employed the topological complexity index [60] as a measure that quantifies the intensity of mixing observed in the motion of multiple navigating agents.”
- Conjugacy in Garside groups I: cyclings, powers and rigidity (2006, background): “As a mapping class group, Bn has structure which, at this time, has not been fully related to its Garside structure, although some interesting relation between the two structures can be found in [19].”