INTERSECTIONS OF MULTICURVES FROM DYNNIKOV COORDINATES



YURTTAŞ, S ÖYKÜ and HALL, TOBY
(2018) INTERSECTIONS OF MULTICURVES FROM DYNNIKOV COORDINATES. Bulletin of the Australian Mathematical Society, 98 (1). pp. 149-158.

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Abstract

<jats:p>We present an algorithm for calculating the geometric intersection number of two multicurves on the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000308_inline1" /><jats:tex-math>$n$</jats:tex-math></jats:alternatives></jats:inline-formula>-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000308_inline2" /><jats:tex-math>$O(m^{2}n^{4})$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000308_inline3" /><jats:tex-math>$m$</jats:tex-math></jats:alternatives></jats:inline-formula>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.</jats:p>

Item Type: Article
Depositing User: Symplectic Admin
Date Deposited: 05 Jun 2018 09:38
Last Modified: 19 Jan 2023 01:32
DOI: 10.1017/s0004972718000308
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3022092