Efficient geodesics and an effective algorithm for distance in the complex of curves
We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.
We would like to thank Ken Bromberg, Chris Leininger, Yair Minsky, Kasra Rafi, and Yoshuke Watanabe for helpful conversations. We are especially grateful to John Hempel for sharing with us his algorithm, to Richard Webb for sharing many ideas and details of his work, and to Tarik Aougab for many insightful comments, especially on the problem of constructing geodesics that are not tight. We would also like to thank Paul Glenn, Kayla Morrell, and Matthew Morse for supplying numerous examples generated by their program Metric in the Curve Complex. Finally, we are grateful to the anonymous referee who made many helpful suggestions.
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