Related papers: Exact Computation for Existence of a Knot Countere…
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…
Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to $24$ crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.
We prove a symmetric version of B\'ezout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective…
This note addresses the meander enumeration problem: "Count all topologically inequivalent configurations of a closed planar non self-intersecting curve crossing a line through a given number of points". We review a description of meanders…
We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
Bezout's theorem gives us the degree of intersection of two properly intersecting projective varieties. As two curves in P^3 never intersect properly, Bezout's theorem cannot be directly used to bound the number of intersection points of…
Due to the orbifold singularities, the intersection numbers on the moduli space of curves $\bar{\sM}_{g,n}$ are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and…
The preservation of ambient isotopic equivalence under piecewise linear (PL) approximation for smooth knots are prominent in molecular modeling and simulation. Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum…
The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the…
It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the…
We suggest a new optical solution for solving the YES/NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then…
The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a…
We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly…
Let $P$ be a polygonal curve in $\mathbb{R}^d$ of length $n$, and $S$ be a point-set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that the Fr\'echet distance from $P$ is less than a…
Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity…
This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The…
This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of…
With an explicit example, we confirm a conjecture by Neumann and Wahl that there exist cusps with no Galois cover by a complete intersection. Some computational techniques are reviewed, and a method for deciding whether a given cusp has a…
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it…