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- W2963578051 abstract "Abstract Przytycki has shown that the size $mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as $|chi(S)|^{k^{2}+k+1}$. In this article, we narrow Przytycki’s bounds, obtaining [ mathcal{N}_{k}(S) =O left( frac{ |chi|^{3k}}{ ( log |chi| )^2 } right)!. ] In particular, the size of a maximal 1-system grows sub-cubically in $|chi(S)|$. The proof uses a circle packing argument of Aougab and Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we use a different argument to show [ mathcal{N}_{k}(S) leq O(n^{2k+2}). ] This may be improved when $k=2$, and we obtain the sharp estimate $mathcal{N}_2(S)=Theta(n^3)$." @default.
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- W2963578051 date "2017-11-13" @default.
- W2963578051 modified "2023-10-17" @default.
- W2963578051 title "Packing Curves on Surfaces with Few Intersections" @default.
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- W2963578051 doi "https://doi.org/10.1093/imrn/rnx270" @default.
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