Some more amplituhedra are contractible

TL;DR

This paper investigates the topological properties of amplituhedra, demonstrating that some are homeomorphic to balls and others are contractible, thereby advancing understanding of their geometric structure.

Contribution

It proves that certain amplituhedra and Grassmann polytopes are homeomorphic to balls or are contractible, extending known topological results.

Findings

Some amplituhedra are homeomorphic to balls

Additional Grassmann polytopes are contractible

Topological complexity of amplituhedra is partially characterized

Abstract

The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int.\ Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.