By Maro A. Natiello, Hernán G Solari
This publication offers the advance and alertness of a few topological tools within the research of information coming from 3D dynamical structures (or comparable objects). the purpose is to stress the scope and obstacles of the equipment, what they supply and what they don't offer. Braid thought, the topology of floor homeomorphisms, information research and the reconstruction of phase-space dynamics are completely addressed.
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Extra info for The user's approach to topological methods in 3-D dynamical systems
Example text
The identification of relevant dynamical properties should occur in a higher level than that of coordinate choice. So we look for orbit properties that persist upon (valid) coordinate transformations, More specifically, we look for properties that persist under homotopies. A homotopy is a continuous transformation that preserves some property that is specified in each particular case. g. a Jordan curve for all t ∈ (0, 1) (what is preserved here is the property of being a closed plane curve without self-intersections).
Let x be a point on a periodic orbit, and W s = {y : d(φt (y), φt (x)) → 0 as t → ∞}. , those points whose dynamical evolution approach the periodic orbit as t → ∞. In simple words it is like the “stable manifold” of the orbit. Clearly, W s (x) will not intersect any other periodic orbit, or its corresponding stable manifold, as two hyperbolic periodic orbits are separated in phase space and no point in phase space can have two different “futures”. Therefore, the map φt is 1-to-1 within the set of periodic orbits of the flow and moreover the collapse preserves each periodic orbit “as it is” and it does not change the linking or knot type.
For example, the linking number between two knots can be obtained adding up the signed crosses between both knots and dividing by two, since each cross adds or subtract a π-turn of one knot over the other. The Conway polynomial can be seen as a bookkeeping of the operations necessary to de-assembling a knot. 2), producing L0 . To each knot or link a polynomial, ∇(z), is assigned, where the polynomial 1 corresponds to the unknot and the relations ∇L+ (z) − ∇L− (z) = −z∇L0 (z) are satisfied. The recursive relation allows to compute the polynomial of the desired knot de-assembling the projection [Carlson 2001].