• Corpus ID: 237213320

Piecewise circular curves and positivity

@inproceedings{Burelle2021PiecewiseCC,
  title={Piecewise circular curves and positivity},
  author={Jean-Philippe Burelle and Ryan Kirk},
  year={2021}
}
We introduce the moduli space of generic piecewise circular n-gons in the Riemann sphere and relate it to a moduli space of Legendrian polygons. We prove that when n = 2k, this moduli space contains a connected component homeomorphic to the Fock-Goncharov space of k-tuples of positive flags for PSp(4,R) and hence is a topological ball. We characterize this component geometrically as the space of simple piecewise circular curves with decreasing curvature. 

Figures and Tables from this paper

References

SHOWING 1-10 OF 18 REFERENCES

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S

Symmetry sets of piecewise-circular curves

  • P. GiblinT. Banchoff
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 1993
Synopsis Piecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various

On the Geometry of Piecewise Circular Curves

In this article we would like to promote a class of plane curves that have a number of special and attractive properties, the piecewise circular curves, or PC curves. (We feel constrained to point

Anosov flows, surface groups and curves in projective space

Note that in [10], W. Goldman gives a complete description of these connected components in the case of finite covers of PSL(2,R). In the case of PSL(2,R), two homeomorphic components, called

Schottky presentations of positive representations

We show that total positivity gives rise to a partial cyclic order on the set of oriented flags in $${\mathbb {R}}^n$$ R n . Using the notion of interval given by this partial cyclic order, we

Symplectic Frieze Patterns

  • S. Morier-Genoud
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2019
We introduce a new class of frieze patterns which is related to symplectic geometry. On the algebraic side, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of

Lie Sphere Geometry

This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of

Total Positivity in Reductive Groups

An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note