490 research outputs found
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
The geometry and combinatorics of cographic toric face rings
In this paper we define and study a ring associated to a graph that we call
the cographic toric face ring, or simply the cographic ring. The cographic ring
is the toric face ring defined by the following equivalent combinatorial
structures of a graph: the cographic arrangement of hyperplanes, the Voronoi
polytope, and the poset of totally cyclic orientations. We describe the
properties of the cographic ring and, in particular, relate the invariants of
the ring to the invariants of the corresponding graph. Our study of the
cographic ring fits into a body of work on describing rings constructed from
graphs. Among the rings that can be constructed from a graph, cographic rings
are particularly interesting because they appear in the study of compactified
Jacobians of nodal curves.Comment: 27 pages; final version, to appear in Algebra & Number Theor
Cluster varieties from Legendrian knots
Many interesting spaces --- including all positroid strata and wild character
varieties --- are moduli of constructible sheaves on a surface with
microsupport in a Legendrian link. We show that the existence of cluster
structures on these spaces may be deduced in a uniform, systematic fashion by
constructing and taking the sheaf quantizations of a set of exact Lagrangian
fillings in correspondence with isotopy representatives whose front projections
have crossings with alternating orientations. It follows in turn that results
in cluster algebra may be used to construct and distinguish exact Lagrangian
fillings of Legendrian links in the standard contact three space.Comment: 47 page
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