55 research outputs found
Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
We continue to develop the tensor-algebra approach to knot polynomials with
the goal to present the story in elementary and comprehensible form. The
previously reviewed description of Khovanov cohomologies for the gauge group of
rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which
are involved rather artificially. We substitute them by alternative and natural
set of cycles, not obligatory planar. Then the whole construction is
straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky
(KR) polynomials, simultaneously for all values of N. No matrix factorization
and related tedious calculations are needed in such approach, which can
therefore become not only conceptually, but also practically useful.Comment: 66 page
A-infinity structure on simplicial complexes
A discrete (finite-difference) analogue of differential forms is considered,
defined on simplicial complexes, including triangulations of continuous
manifolds. Various operations are explicitly defined on these forms, including
exterior derivative and exterior product. The latter one is non-associative.
Instead, as anticipated, it is a part of non-trivial A-infinity structure,
involving a chain of poly-linear operations, constrained by nilpotency
relation: (d + \wedge + m + ...)^n = 0 with n=2.Comment: final version. 29 page
On the shapes of elementary domains or why Mandelbrot Set is made from almost ideal circles?
Direct look at the celebrated "chaotic" Mandelbrot Set in Fig..\ref{Mand2}
immediately reveals that it is a collection of almost ideal circles and
cardioids, unified in a specific {\it forest} structure. In /hep-th/9501235 a
systematic algebro-geometric approach was developed to the study of generic
Mandelbrot sets, but emergency of nearly ideal circles in the special case of
the family was not fully explained. In the present paper the shape of
the elementary constituents of Mandelbrot Set is explicitly {\it calculated},
and difference between the shapes of {\it root} and {\it descendant} domains
(cardioids and circles respectively) is explained. Such qualitative difference
persists for all other Mandelbrot sets: descendant domains always have one less
cusp than the root ones. Details of the phase transition between different
Mandelbrot sets are explicitly demonstrated, including overlaps between
elementary domains and dynamics of attraction/repulsion regions. Explicit
examples of 3-dimensional sections of Universal Mandelbrot Set are given. Also
a systematic small-size approximation is developed for evaluation of various
Feigenbaum indices.Comment: 65 pages, 30 figure
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