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On the Moduli Space of a Spherical Polygonal Linkage

Published online by Cambridge University Press:  20 November 2018

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Abstract

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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 $\left( n\,-\,1 \right)$ side-lengths are fixed. We obtain a Morse function on the $\left( n\,-\,2 \right)$-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.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1999

References

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