28,061 research outputs found
Integral geometry of plane curves and knot invariants
We study the integral expression of a knot invariant obtained as the second
coefficient in the perturbative expansion of Witten's Chern-Simons path
integral associated with a knot. One of the integrals involved turns out to be
a generalization of the classical Crofton integral on convex plane curves and
it is related with invariants of generic plane curves defined by Arnold
recently with deep motivations in symplectic and contact geometry. Quadratic
bounds on these plane curve invariants are derived using their relationship
with the knot invariant.Comment: 18 pages, amslatex, 8 figures not included (will send upon request
The invariant tori of knot type and the interlinked invariant tori in the Nos\'e-Hoover system
We revisit the famous Nos\'e-Hoover system in this paper and show the
existence of some averagely conservative regions which are filled with an
infinite sequence of nested tori. Depending on initial conditions, some
invariant tori are of trefoil knot type, while the others are of trivial knot
type. Moreover, we present a variety of interlinked invariant tori whose
initial conditions are chosen from different averagely conservative regions and
give all the interlinking numbers of those interlinked tori, showing that this
quadratic system possesses so rich dynamic properties.Comment: 8 pages,8 figure
- …