Searching for non-order-preserving braids algorithmically

2024/10/14 by Johnson, Jonathan, Scherich, Nancy, Turner, Hannah
#57K10 #57K20 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2410.10595

Abstract

An n-strand braid is order-preserving if its action on the free group F_n preserves some bi-order of F_n. A braid β is order-preserving if and only if the link L obtained as the union of the closure of β and its axis has bi-orderable complement. We describe and implement an algorithm which, given a non-order-preserving braid β, confirms this property and returns a proof that β is indeed not order-preserving. Guided by the algorithm, we prove that the infinite family of simple 3-braids σ₁σ₂^2m+1 are not order-preserving for any integer m.

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