We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse knots with one maximum that distinguishes two different representations of the figure eight knot. We also present the results of computer calculations for some i...

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. introduce twisted cartesian product two algebras according to linear representations by infinitesimal transformations. also exhibit double extension process algebra which allows us get all simply connected groups with bi-invariant symplectic connections. All constructed performin...

For a topological dynamical system (X, f), consisting of a continuous map f : X → X, and a (not necessarily compact) set Z ⊂ X, Bowen [2] defined a dimension-like version of entropy, hX(f, Z). In the same work, he introduced a notion of entropyconjugacy for pairs of invertible compact systems: the systems (X, f) and (Y, g) are entropy-conjugate if there exist invariant Borel sets X ′ ⊂ X and Y ...

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associa...

We consider a dynamical system whose phase space contains a two-dimensional normally hyperbolic invariant manifold diffeomorphic to an annulus. We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map. We assume that in the annulus there exist finite sequences of primary invariant Lipschitz tori of dimension 1, with the property that the unstable m...

There is a well known one–parameter family of left invariant CR structures on SU(2) ∼= S. We show how purely algebraic methods can be used to explicitly compute the canonical Cartan connections associated to these structures and their curvatures. We also obtain explicit descriptions of tractor bundles and tractor connections.

We show that the groups of finite energy loops and paths (that is, those Sobolev class H 1 ) with values in a compact connected Lie group, as well their central extensions, satisfy an amenability-like property: they admit left-invariant mean on space bounded functions uniformly continuous regard to metric. Every strongly unitary representation π such group (which we call skew-amenable) has conj...