Related papers: Two-variable Conway polynomial and Cochran's deriv…

In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ which is a rational function in variables $a$ and $s$ and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this…

It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…

Let $2 \leq y \leq x$ such that $\beta := \frac{\log x}{\log y} \rightarrow \infty$. Let $\omega_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $\mu_y(n) := \mu^2(n)(-1)^{\omega_y(n)}$, where…

We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We…

We consider a polynomial $P\in \mathbb{R}[x_{1},\cdots, x_{d}]$ of degree $ \delta $ that depends non-trivially on each of $x_1,...,x_d$ with $d\geq 2$. For any integer $t$ with $2\leq t\leq d$, any natural number $n \in \mathbb{N}$, and…

Let $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with $X$. On one hand, the Kailath--Segall formula gives the relationship between…

We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are…

We give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann about graph links.

Representations of polynomial covariance type commutation relations are constructed on Banach spaces $L_p$ and $C[\alpha, \beta],\ \alpha,\beta\in \mathbb{R}$. Representations involve operators with piecewise functions, multiplication…

We review the framework subfactors provide for understanding modular invariants. We discuss the structure of a generalized Longo-Rehren subfactor and the relationship between the coupling matrices of such subfactors, modular invariance and…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as…

We define a knot/link invariant using set theoretical solutions $(X,\sigma)$ of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given $(X,\sigma)$, a universal group Unc(X) governing all 2-cocycles in $X$, and…

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L),…

Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We…

Let $L$ be a oriented link such that $\Sigma_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander…

Let $L$ be a linear symmetric differential operators on $L^{2}\left( \mathbb{R}\right) $ whose domain is the Schwartz test function space, $\mathcal{S}.$ For the majority of this paper, it is assumed that the coefficient of $L$ are…

We compute the nonabelian $\mathrm{SL_2}(\mathbb{C})$-character varieties of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m$ and $n$ are non-zero integers. By studying real points on these varieties, we determine the…

We examine a two-parameter ($\hbar ,$ $\lambda $) deformation of the Poincar\`e algebra which is covariant under the action of $SL_q(2,C).$ When $\lambda \rightarrow 0$ it yields the Poincar\`e algebra, while in the $\hbar\rightarrow 0$…

We show that any semi-direct sum $L$ of Lie algebras with Levi factor $S$ must be perfect if the representation associated with it does not possess a copy of the trivial representation. As a consequence, all invariant functions of $L$ must…