Related papers: Kauffman-Jones polynomial of a curve on a surface

Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partial\Sigma\neq\varnothing$ and punctures $\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. In this paper we show that for every curve $\gamma$…

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $\gamma$ be a simple closed curve on $M$. Let $m \gamma$ denote the curve formed by tracing $\gamma$ $m$ times. We prove that if $m \gamma$ is contractible through…

Let $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ be a point with algebraically conjugate coordinates such that the…

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the…

We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson…

Suppose a smooth planar curve $\gamma$ is $2\pi$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $\gamma$ self-intersects, then it has a segment of length $\ell- 2\pi$ on which it self-intersects…

We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlev\'e equations, in a Lax form,…

Given any $\Gamma=\gamma(\mathbb{S}^1)\subset\mathbb{R}^2$, image of a Lipschitz curve $\gamma:\mathbb{S}^1\rightarrow \mathbb{R}^2$, not necessarily injective, we provide an explicit formula for computing the value of \[ \mathcal…

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…

We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$.…

Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number…

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-\tau^2$ with $\tau>0$. Let $N(\phi)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the…

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1, x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $\gamma$ is a…

The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the…

A curve \gamma in the plane is t-monotone if its interior has at most t-1 vertical tangent points. A family of t-monotone curves F is \emph{simple} if any two members intersect at most once. It is shown that if F is a simple family of n…

In this paper, the $L^2$ boundedness of the Hilbert transform along variable flat curve $(t,P(x_1)\gamma(t))$ $$H_{P,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-P(x_1)\gamma(t))\,\frac{\textrm{d}t}{t},\quad…

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation…

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating…