Related papers: Conjectured enumeration of Vassiliev invariants

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be…

We deal with $r$-regular bipartite graphs with $2n$ vertices. In a previous paper, Butera, Pernici and the author have introduced a quantity $u(i)$, $u(i) = -\ln(i!m\_i)$, a function of the number of $i$-matchings, $m\_i$, and conjectured…

Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all…

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…

We develop invariant theory for the quantum group ${\rm U}_q$ of $G_2$ at generic $q$ in the setting of braided symmetric algebras. Let ${\mathcal A}_m$ be the braided symmetric algebra over $m$-copies of the $7$-dimensional simple ${\rm…

Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…

A first characterization of the isomorphism classes of $k$-involutions for any reductive algebraic group defined over a perfect field was given in \cite{Helm2000} using $3$ invariants. In \cite{HWD04,Helm-Wu2002} a full classification of…

In this paper, we construct and analyze Bessel and Flett potentials associated with the heat and Poisson semigroups in the framework of the $(k,1)$-generalized Fourier transform. We establish fundamental properties of these potentials and…

Functions satisfying the functional equation \begin{align*} \sum_{r=0}^{n-1} (-1)^r f(x+ry, ny) = f(x,y), \quad \text{for any positive odd integer $n$}, \end{align*} are named the alternating invariant functions. Examples of such functions…

We calculate the knot invariant coming from the Teichm\"{u}ller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichm\"{u}ller TQFT…

Vogel's universality gives a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha,\beta,\gamma$, which are homogeneous coordinates of Vogel's plane. It is associated…

Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows…

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$ where the sequence $x = (x(n))_{n \in…

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically…

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…

We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments…

For a given $(X,S,\beta)$, where $S,\beta\colon X\times X\to X\times X$ are set theoretical solutions of Yang-Baxter equation with a compatibility condition, we define an invariant for virtual (or classical) knots/links using non…

We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and…

Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…

We conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov-Witten theory of Calabi-Yau fivefolds with a torus action. We prove the conjecture for skeletal, locally anti-diagonal torus actions by…