286 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
Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
An elementary introduction to Khovanov construction of superpolynomials.
Despite its technical complexity, this method remains the only source of a
definition of superpolynomials from the first principles and therefore is
important for development and testing of alternative approaches. In this first
part of the review series we concentrate on the most transparent and
unambiguous part of the story: the unreduced Jones superpolynomials in the
fundamental representation and consider the 2-strand braids as the main
example. Already for the 5_1 knot the unreduced superpolynomial contains more
items than the ordinary Jones.Comment: 33 page
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