Abstract
We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure 3-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all 3-braids. We also obtain effective upper and lower bounds for the entropy of pure 3-braids in these terms. The proof leads to the study of the extremal length (in the sense of Ahlfors) of classes of curves representing elements of the fundamental group of the twice punctured complex plane.
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Acknowledgements
The author is grateful to the SFB “Raum-Zeit-Materie” at Humboldt-University Berlin for support in the beginning of the work on the paper, and to the Max-Planck-Institute in Bonn where the main part of the work was done. O.Viro stimulated the work asking about an invariant of braids rather than invariants of conjugacy classes of braids (like entropy and conformal module of conjugacy classes). The author would like to thank Alexander Weiße for teaching how to draw figures and for producing the wonderful essential parts of the difficult figures.
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Jöricke, B. Fundamental groups, 3-braids, and effective estimates of invariants. Math. Z. 294, 1553–1609 (2020). https://doi.org/10.1007/s00209-019-02317-6
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DOI: https://doi.org/10.1007/s00209-019-02317-6