3,725 research outputs found
Hurwitz equivalence of braid monodromies and extremal elliptic surfaces
We discuss the equivalence between the categories of certain ribbon graphs
and subgroups of the modular group and use it to construct
exponentially large families of not Hurwitz equivalent simple braid monodromy
factorizations of the same element. As an application, we also obtain
exponentially large families of {\it topologically} distinct algebraic objects
such as extremal elliptic surfaces, real trigonal curves, and real elliptic
surfaces
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.
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
Matrix factorizations and link homology
For each positive integer n the HOMFLY polynomial of links specializes to a
one-variable polynomial that can be recovered from the representation theory of
quantum sl(n). For each such n we build a doubly-graded homology theory of
links with this polynomial as the Euler characteristic. The core of our
construction utilizes the theory of matrix factorizations, which provide a
linear algebra description of maximal Cohen-Macaulay modules on isolated
hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep
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