Oriented Area as a Morse Function on Configuration Spaces of Necklaces

Daniil Mamaev

Published 2020-01-08Version 1

Assume one has a necklace, that is, a closed string with a number of beads on it. Some of the beads are fixed, some can slide along the string. For a planar configuration of a necklace, the oriented area is well-defined. We characterise critical points of the oriented area function in geometric terms and give a formula for the Morse indices. Thus we obtain a generalisation of Jacob Steiner's isoperimetric theorems for polygons in the plane.