Homotopy Type of Intervals of the Second Higher Bruhat Orders

Abstract

The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is a poset on commutation classes of reduced words for the longest element of a type A Coxeter system. For this case, we prove that the noncontractible intervals are in natural correspondence with the zonogonal tilings of a zonogon. Our proof uses some tools developed by Felsner and Weil to show that the two standard orderings on the rhombic tilings of a zonogon are identical. arXiv:1510.03493v1 [math.CO]

Algebraic Weaves and Braid Varieties

“…In the spherical case, we conjecture to always have such polytopal realizations. Note that McConville [67] proved that intervals in the second higher Bruhat orders are contractible or homotopy equivalent to spheres.…”

“…Remark 4.18. The polytopes in [67] are the Hasse graphs of second higher Bruhat orders introduced by Manin and Shekhtman [65,66], see also [90]. Given an arbitrary braid β, we can consider a similar oriented graph D β .…”