Related papers: Dressing curves

In this paper we provide a characterization for a class of convex curves on the 3-sphere. More precisely, using a theorem that decomposes a locally convex curve on the 3-sphere as a pair of curves on the 2-sphere, one of which is locally…

We revisit the group structure on elliptic curves and give a simple and elementary proof of the associativity of the addition. We do this by providing an explicit formula for the sum of three points, only using the explicit definition of…

We study the distribution of the individual components of a random multicurve under the action of the mapping class group.

The distribution of degree $d$ points on curves is well understood, especially for low degrees. We refine this study to include information on the Galois group in the simplest interesting case: $d = 3$. For curves of genus at least 5, we…

We describe a simple algorithm for classifying orbits into orbit families. This algorithm works by finding patterns in the sign changes of the principal coordinates. Orbits in the logarithmic potential are studied as an application; we…

We classify simple parametrisations of complex curve singularities. Simple means that all neighbouring singularities fall in finitely many equivalence classes. We take the neighbouring singularities to be the ones occurring in the versal…

This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…

We give an elementary proof of the group law for elliptic curves using explicit formulas.

The notion of constant cycle curves on K3 surfaces is introduced. These are curves that do not contribute to the Chow group of the ambient K3 surface. Rational curves are the most prominent examples. We show that constant cycle curves…

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

In this paper we compute the number of rational curves with one node passing through a given number of points, lines and tangent to a given number of planes in $\mathbb{P}^3$.

We characterize the curves in $ P^3$ which are minimal in their biliaison class. Such curves are exactly the curves which do no admit an elementary descending biliaison. As a consequence we have that every curve in $P^3$ can be obtained…

We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.

We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

The loop braid group is the motion group of unknotted oriented circles in $\mathbb{R}^3$. In this paper, we study their representations through the approach inspired by two dimensional topological phases of matter. In principle, the motion…

We characterise when a set of simple closed curves in an orientable surface forms a bouquet, in terms of relations between the corresponding Dehn twists.

In this article, we give a numerical algorithm to compute braid groups of curves, hyperplane arrangements, and parameterized system of polynomial equations. Our main result is an algorithm that determines the cross-locus and the generators…

We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not…

In this article, circular arcs are considered both individually and as elements of a piecewise circular curve. The endpoint parameterization proves to be quite advantageous here. The perspective of symplectic geometry provides new vectorial…