Geometric intersection of curves on non--orientable genus two surfaces

We present formulae for calculating the geometric intersection number of an arbitrary multicurve with so--called elementary curves on non--orientable surfaces of genus $2$, with finitely many punctures and one boundary component, making use of generalized Dynnikov coordinates.Comment: 16 pages, 11 figures; proof of Theorem 1 and Remarks 1 update

INTERSECTIONS OF MULTICURVES FROM DYNNIKOV COORDINATES

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.</jats:p

Curves on Non-Orientable Surfaces and Crosscap Transpositions

Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston&rsquo;s theory of surface homeomorphisms, which were solved only on orientable surfaces before

Anatomy of the insular veins

[No abstract available