Related papers: Link homology and categorification
The purpose of this note is to extend to Brownian loops some homology and holonomy results obtained in the case of discrete loops on a graph
We present an unified construction for algebras and modules homologies and cohomologies, in the case of associative, commuttaive, Lie and Gerstenhaber algebras. We make a distinction between the linear part of the construction of algebras…
We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions.…
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer…
In this paper, for each graph $G$, we def\mbox{}ine a chain complex of graded modules over the ring of polynomials, whose graded Euler characteristic is equal to the chromatic polynomial of $G$. Furthermore, we def\mbox{}ine a chain complex…
We define and study the properties of observables associated to any link in $\Sigma\times {\bf R}$ (where $\Sigma$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces…
We define the notion of a trace kernel on a manifold M. Roughly speaking, it is a sheaf on M x M for which the formalism of Hochschild homology applies. We associate a microlocal Euler class to such a kernel, a cohomology class with values…
We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To…
We describe polar homology groups for complex manifolds. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue…
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov…
Following the approach to grid homology of links in $S^3$, we prove combinatorially that the grid homology of links in lens spaces defined by Baker, Grigsby, and Hedden is a link invariant. Further, using the sign assignment defined by…
Although most knots are nonalternating, modern research in knot theory seems to focus on alternating knots. We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones…
We give a congruence relating a one variable specialization of the two variable Kauffman polynomial of any periodic link to that of its mirror image. Consequently, we obtain a new and simple criterion for periodicity of links.
This paper defines the Euler-Poincar\'{e} characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions;…
Two links are called link-homotopic if they are transformed to each other by a sequence of self-crossing changes and ambient isotopies. The notion of link-homotopy is generalized to spatial graphs and it is called component-homotopy. The…
Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…
We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…
This is a survey of various types of Floer theories (both in symplectic geometry and gauge theory) and relations among them.