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Moduli Space of a Planar Polygonal Linkage: A Combinatorial Description

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Abstract

We describe and study an explicit structure of a regular cell complex \(\mathcal {K}(L)\) on the moduli space M(L) of a planar polygonal linkage L. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space M is a sphere, the complex \(\mathcal {K}\) is dual to the boundary complex of the permutohedron.The dual complex \(\mathcal {K}^*\) is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron.

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Acknowledgements

I am grateful to Nikolai Mnev for inspiring conversations. I am also indebted to Misha Kapovich for delivering me the proof of Lemma 1.2.

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  1. St. Petersburg Department of V.A. Steklov Institute of Mathematics RAS, St. Petersburg State University, Saint Petersburg, Russia

    Gaiane Panina

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  1. Gaiane Panina
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Correspondence to Gaiane Panina.

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Panina, G. Moduli Space of a Planar Polygonal Linkage: A Combinatorial Description. Arnold Math J. 3, 351–364 (2017). https://doi.org/10.1007/s40598-017-0070-1

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  • DOI: https://doi.org/10.1007/s40598-017-0070-1

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