Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-14T01:26:57.890Z Has data issue: false hasContentIssue false
  • English
  • Français

INTERSECTIONS OF MULTICURVES FROM DYNNIKOV COORDINATES

Part of: Special aspects of infinite or finite groups Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  03 May 2018

S. ÖYKÜ YURTTAŞ  and
TOBY HALL
S. ÖYKÜ YURTTAŞ
Affiliation:
Dicle University, Science Faculty, Mathematics Department, 21280, Diyarbakır, Turkey email [email protected]
TOBY HALL*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an algorithm for calculating the geometric intersection number of two multicurves on the $n$-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity $O(m^{2}n^{4})$, where $m$ is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredient is an algorithm due to Cumplido for relaxing a multicurve.

Keywords

geometric intersectionmulticurvespunctured diskDynnikov coordinates

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 20F36: Braid groups; Artin groups 57N16: Geometric structures on manifolds
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 149 - 158
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Artin, E., ‘Theory of braids’, Ann. of Math. (2) 48 (1947), 101–126.Google Scholar
Bell, M. and Webb, R., ‘Applications of fast triangulation simplification’, Preprint, 2016, arXiv:1605.03514.Google Scholar
Birman, J. S., Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, 82 (Princeton University Press, Princeton, NJ, 1974).Google Scholar
Cumplido, M., ‘On the minimal positive standardizer of a parabolic subgroup of an Artin–Tits group’, Preprint, 2017, arXiv:1708.09310.Google Scholar
Dehornoy, P., Dynnikov, I., Rolfsen, D. and Wiest, B., Ordering Braids, Mathematical Surveys and Monographs, 148 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Dynnikov, I., ‘On a Yang-Baxter mapping and the Dehornoy ordering’, Uspekhi Mat. Nauk 57(3(345)) (2002), 151–152.Google Scholar
Hall, T. and Yurttaş, S. Ö., ‘On the topological entropy of families of braids’, Topol. Appl. 156(8) (2009), 1554–1564.CrossRefGoogle Scholar
Moussafir, J.-O., ‘On computing the entropy of braids’, Funct. Anal. Other Math. 1(1) (2006), 37–46.Google Scholar
Schaefer, M., Sedgwick, E. and Štefankovič, D., ‘Computing Dehn twists and geometric intersection numbers in polynomial time’, in: Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG2008) (2008), 111–114.Google Scholar
Yurttaş, S. Ö., ‘Geometric intersection of curves on punctured disks’, J. Math. Soc. Japan 65(4) (2013), 1153–1168.Google Scholar