2,696 research outputs found
The foam and the matrix factorization sl3 link homologies are equivalent
We prove that the foam and matrix factorization universal rational sl3 link
homologies are naturally isomorphic as projective functors from the category of
link and link cobordisms to the category of bigraded vector spaces.Comment: We have filled a gap in the proof of Lemma 5.2. 28 page
A note on Khovanov-Rozansky -homology and ordinary Khovanov homology
In this note we present an explicit isomorphism between Khovanov-Rozansky
-homology and ordinary Khovanov homology. This result was originally
stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet
to appear in the literature. The main missing detail is providing a coherent
choice of signs when identifying variables in the -homology. Along with
the behavior of the signs and local orientations in the -homology, both
theories behave differently when we try to extend their definitions to virtual
links, which seemed to suggest that the -homology may instead correspond
to a different variant of Khovanov homology. In this paper we describe both
theories and prove that they are in fact isomorphic by showing that a coherent
choice of signs can be made. In doing so we emphasize the interpretation of the
-complex as a cube of resolutions.Comment: 19 pages, 11 figures. Expanded introduction and abstract. Remark
added to end of section 4.
Tame Decompositions and Collisions
A univariate polynomial f over a field is decomposable if f = g o h = g(h)
for nonlinear polynomials g and h. It is intuitively clear that the
decomposable polynomials form a small minority among all polynomials over a
finite field. The tame case, where the characteristic p of Fq does not divide n
= deg f, is fairly well-understood, and we have reasonable bounds on the number
of decomposables of degree n. Nevertheless, no exact formula is known if
has more than two prime factors. In order to count the decomposables, one wants
to know, under a suitable normalization, the number of collisions, where
essentially different (g, h) yield the same f. In the tame case, Ritt's Second
Theorem classifies all 2-collisions.
We introduce a normal form for multi-collisions of decompositions of
arbitrary length with exact description of the (non)uniqueness of the
parameters. We obtain an efficiently computable formula for the exact number of
such collisions at degree n over a finite field of characteristic coprime to p.
This leads to an algorithm for the exact number of decomposable polynomials at
degree n over a finite field Fq in the tame case
Deformations of colored sl(N) link homologies via foams
We generalize results of Lee, Gornik and Wu on the structure of deformed
colored sl(N) link homologies to the case of non-generic deformations. To this
end, we use foam technology to give a completely combinatorial construction of
Wu's deformed colored sl(N) link homologies. By studying the underlying
deformed higher representation theoretic structures and generalizing the
Karoubi envelope approach of Bar-Natan and Morrison we explicitly compute the
deformed invariants in terms of undeformed type A link homologies of lower rank
and color.Comment: 64 pages, many figure
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