Arcs Intersecting at Most Once
نویسنده
چکیده
We prove that on a punctured oriented surface with Euler characteristic χ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most once is 2|χ|(|χ|+1). This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |χ| for sets of arcs and curves pairwise intersecting a uniformly bounded number of times. Finally, we prove that on a punctured sphere the maximal cardinality of a set of arcs starting and ending at specified punctures and pairwise intersecting at most once is 12 |χ|(|χ|+ 1).
منابع مشابه
Arcs on Spheres Intersecting at Most Twice
Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set A of arcs pairwise intersecting at most once, which start at p and end in Q, is |χ|(|χ| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise intersecting at most twice is |χ|(|χ|+ 1)(|χ|+ 2).
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