15,853 research outputs found
Integrability of the Pentagram Map
The pentagram map was introduced by R. Schwartz in 1992 for convex planar
polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved
Liouville integrability of the pentagram map for generic monodromies by
providing a Poisson structure and the sufficient number of integrals in
involution on the space of twisted polygons. In this paper we prove
algebraic-geometric integrability for any monodromy, i.e., for both twisted and
closed polygons. For that purpose we show that the pentagram map can be written
as a discrete zero-curvature equation with a spectral parameter, study the
corresponding spectral curve, and the dynamics on its Jacobian. We also prove
that on the symplectic leaves Poisson brackets discovered for twisted polygons
coincide with the symplectic structure obtained from Krichever-Phong's
universal formula.Comment: 33 pages, 1 figure; v3: substantially revise
Involutes of Polygons of Constant Width in Minkowski Planes
Consider a convex polygon P in the plane, and denote by U a homothetical copy
of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a
norm such that, with respect to this norm, P has constant Minkowskian width. We
define notions like Minkowskian curvature, evolutes and involutes for polygons
of constant U-width, and we prove that many properties of the smooth case,
which is already completely studied, are preserved. The iteration of involutes
generates a pair of sequences of polygons of constant width with respect to the
Minkowski norm and its dual norm, respectively. We prove that these sequences
are converging to symmetric polygons with the same center, which can be
regarded as a central point of the polygon P.Comment: 20 pages, 11 figure
Minimal knotted polygons in cubic lattices
An implementation of BFACF-style algorithms on knotted polygons in the simple
cubic, face centered cubic and body centered cubic lattice is used to estimate
the statistics and writhe of minimal length knotted polygons in each of the
lattices. Data are collected and analysed on minimal length knotted polygons,
their entropy, and their lattice curvature and writhe
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