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# On the Moduli Space of a Spherical Polygonal Linkage

Published:1999-09-01
Printed: Sep 1999
• Michael Kapovich
• John J. Millson
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## Abstract

We give a wall-crossing'' formula for computing the topology of the moduli space of a closed $n$-gon linkage on $\mathbb{S}^2$. We do this by determining the Morse theory of the function $\rho_n$ on the moduli space of $n$-gon linkages which is given by the length of the last side---the length of the last side is allowed to vary, the first $(n - 1)$ side-lengths are fixed. We obtain a Morse function on the $(n - 2)$-torus with level sets moduli spaces of $n$-gon linkages. The critical points of $\rho_n$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of $\rho_n$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.
 MSC Classifications: 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14P05 - Real algebraic sets [See also 12Dxx, 13P30]

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