Fundamental Groups, 3-Braids, and Effective Estimates of Invariants

Burlind Joricke

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arXiv · 1712.10307

Fundamental Groups, 3-Braids, and Effective Estimates of Invariants

Burlind Joricke

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Abstract

We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-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 three-braids. We also obtain effective upper and lower bounds for the entropy of pure three-braids in these terms. The proof leads to the study of the extremal length of classes of curves representing elements of the fundamental group of the twice punctured complex plane.

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Burlind Joricke. 2017-12-29. Fundamental Groups, 3-Braids, and Effective Estimates of Invariants. https://arxiv.org/abs/1712.10307

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