Categorification of knot and graph polynomials and the polynomial ring.

摘要

Categorification lifts numbers to vector spaces and vector spaces to categories. A prime example is turning Euler characteristic of a topological space into its homology groups. Important examples include various link homology groups which lift polynomial invariants of knots. For instance, Khovanov homology lifts the Jones polynomial and Ozsvath-Szabo-Rassmussen homology lifts the Alexander polynomial. Knots, via their diagrams, are closely related to planar graphs. Recently, several graph invariants have also been categorified, such as the chromatic and the Tutte polynomial. Chromatic cohomology for graphs provides a link between link homology and well developed theory of Hochschild homology [Pr2]. We utilize this correspondence to partially prove A. Shumakovitch's conjecture claiming that alternating links can have only 2-torsion and that any link which is not a connected or disjoint sum of Hopf links and trivial links has torsion in Khovanov homology [Sh]. Shumakovitch proved the conjecture for alternating links and M. Asaeda, J. Przytycki [AP] proved the existence of Z2-torsion in Khovanov homology of a large class of adequate links. We extend these results using the modified chromatic graph cohomology to semi-adequate knots [PS], and we explicitly compute 2-torsion [PPS]. These results can be used for describing torsion in Khovanov homology of some positive braids. Moreover, we analyze torsion in chromatic cohomology over other algebras of truncated polynomials, focusing on algebra Z [x]/x3.;Inspired by the general idea of categorification, introduced by Mikhail Khovanov, we construct categorification of one variable polynomials [KS]. This ring becomes a Grothendieck ring of a suitable additive monoidal category of (finitely generated) projective modules over an idempotented geometrically defined ring. Monomials xn become indecomposable projective modules, while polynomials (x -- 1) n turn into so-called standard modules. These collections of modules together with simple modules satisfy the Bernstein-Gelfand-Gelfand reciprocity property. Various basic structures of the theory of orthogonal polynomials, such as the kernels that approximate the identity operator, admit categorical lifting in our framework, which generalizes to the categorification of one variable orthogonal polynomials, including Chebyshev and Hermite polynomials.