相关论文: On the LMO Invariant, the Wheeling Theorem, and th…

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of Chinese characters. The…

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the…

We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and…

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental…

This work develops some technology for accessing the loop expansion of the Kontsevich integral of a knot. The setting is an application of the LMO invariant to certain surgery presentations of knots by framed links in the solid torus. A…

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories…

Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which…

We propose an abelian categorification of $\hat{Z}$-invariants for Seifert $3$-manifolds. First, we give a recursive combinatorial derivation of these $\hat{Z}$-invariants using graphs with certain hypercubic structures. Next, we consider…

We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only…

The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…

In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface,…

We relate the tree part of the Aarhus integral to Milnor's mu-invariants of string-links in homology balls thus generalizing results of Habegger and Masbaum.

The Labastida-Marin\~o-Ooguri-Vafa (LMOV) invariants are the open string BPS invariants which are expected to be integers based on the string duality conjecture from M-theory. Several explicit formulae of LMOV invariants for framed unknot…

Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based…

A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on…

We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied…

This paper investigates the algebraic and dynamical properties of the twisted cocycle, a $\mathrm{GL}(d, \mathbb{C})$-valued cocycle defined over the toral extension of the Zorich (Rauzy-Veech) renormalization for interval exchange…

Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times…