3,725 research outputs found

    Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

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    We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group Γ\Gamma 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 sl2sl_2-homology and ordinary Khovanov homology

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    In this note we present an explicit isomorphism between Khovanov-Rozansky sl2sl_2-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 sl2sl_2-homology. Along with the behavior of the signs and local orientations in the sl2sl_2-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the sl2sl_2-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 sl2sl_2-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

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    The Hamilton-Waterloo Problem HWP(v;m,n;α,β)(v;m,n;\alpha,\beta) asks for a 2-factorization of the complete graph KvK_v or KvIK_v-I, the complete graph with the edges of a 1-factor removed, into α\alpha CmC_m-factors and β\beta CnC_n-factors, where 3m<n3 \leq m < n. In the case that mm and nn are both even, the problem has been solved except possibly when 1{α,β}1 \in \{\alpha,\beta\} or when α\alpha and β\beta are both odd, in which case necessarily v2(mod4)v \equiv 2 \pmod{4}. 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(v;2m,2n;α,β)(v;2m,2n;\alpha,\beta) for odd α\alpha and β\beta whenever the obvious necessary conditions hold, except possibly if β=1\beta=1; β=3\beta=3 and gcd(m,n)=1\gcd(m,n)=1; α=1\alpha=1; or v=2mn/gcd(m,n)v=2mn/\gcd(m,n). This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above

    Matrix factorizations and link homology

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    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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