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 and Trnka (J High Energy Phys 10:30, 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 (in: Jerison, Kisin, Seidel, Stanley, Yau, Yau (eds) Current developments in mathematics, International Press, Somerville, 2016). In this paper we show that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.
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Acknowledgements
The authors thank Rainer Sinn for sharing the knowledge about semi-algebraic sets, to Thomas Lam, whose great observations increased the generality of the results in this paper, and to Steven Karp for helpful comments. We are grateful to the referee for careful reading of our manuscript and for useful suggestions that improved the quality of our paper.
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The research by Pavle V. M. Blagojević has received funding from the Grant ON 174024 of the Serbian Ministry of Education and Science. The research by Nevena Palić has received funding from DFG via the Berlin Mathematical School. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 during the Fall of 2017, while all authors were in residence at the Mathematical Sciences Research Institute in Berkeley CA.
