Related papers: Some knot theory of complex plane curves
We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…
We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc…
In this diploma thesis we discuss the deformation theory of Lie algebroids and Dirac structures. The first chapter gives a short introduction to Dirac structures on manifolds as introduced by Courant in 1990. We also give some physical…
Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that $C^2$ closed convex curves possess at least two tripod configurations. Later, Kao and…
In this paper, we give some extension of fundamental theorems in Nevanlinna - Cartan theory for holomorphic curve on M punctured complex planes. As an application, we establish a result for uniqueness problem of holomorphic curve by inverse…
In this preprint we present an outline of the multidimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvability by radicals, by…
It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two…
Real quadric curves are often referred to as "conic sections," implying that they can be realized as plane sections of circular cones. However, it seems that the details of this equivalence have been partially forgotten by the mathematical…
In a previous paper (q-alg/9501022) we suggested some algorithms that could be useful in solving the problem of knot classification. Here we continue this discussion by answering questions raised in that paper and by commenting on practical…
In the paper Factorisation of division polynomials (H. Verdure, Proc. japan Academy, Ser A. 80 n. 5), Verdure gives the factorisation patterns of division polynomials of elliptic curves defined over a finite field. However, the result given…
In the paper published in Duke Math. J. 1993, Y. Wen studied a second-order parabolic equation for inextensible elastic \emph{closed} curves in $\mathbb{R}^{2}$ toward inextensible elasticae. In this article, we extend Wen's result to the…
This short article, which appeared in the MSRI Fall 2004 newletter, is intended for a general mathematical audience. It describes some mathematical themes from the Winter 2004 program at MSRI in topological aspects of real algebraic…
We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin…
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood…
We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been…
We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…
These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A…
This paper is based on my talks (`Skein modules with a cubic skein relation: properties and speculations' and `Symplectic structure on colorings, Lagrangian tangles and its applications') given in Kyoto (RIMS), September 11 and September 18…
The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches…