Related papers: Exact Computation for Existence of a Knot Countere…
Let $D$ be a knot diagram, and let ${\mathcal D}$ denote the set of diagrams that can be obtained from $D$ by crossing exchanges. If $D$ has $n$ crossings, then ${\mathcal D}$ consists of $2^n$ diagrams. A folklore argument shows that at…
We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of $n$-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of…
A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each…
In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon $\mathcal P$ and an integer $k$, and the question is if there exist $k$ convex polygons whose union is $\mathcal P$. It is known that MCC is $\mathsf{NP}$-hard…
We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the…
The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in…
We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is…
One of the goals of this paper is to prove that the index of intersection of two complex curves in a two-dimensional complex manifold tangent to each other at a common boundary point is positive. This is achieved via the construction of a…
We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on…
Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree $d\ge 2$, we provide lower bounds for the following four numerical invariants:…
Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.
This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly $1$. This immediately implies a positive answer to a…
Consider the collection of edge bicolorings of a graph that is cellularly embedded on an orientable surface. In this work, we count the number of equivalence classes of such colorings under two relations: reversing colors around a face and…
We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a…
Several authors have recently attempted to show that the intersection of three simply connected subcontinua of the plane is simply connected provided it is non-empty and the intersection of each two of the continua is path connected. In…
Let c(K;F) denote the surface crossing number of a knot K with respect to a closed connected surface F in S^3. We relate c(K;F) to the tunnel number t(K) and to the Heegaard deficiency delta(F)=g(M_1;F)+g(M_2;F)-g(F), where S^3=M_1 union_F…
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many…
Numerical simulations indicate that there exist conformations of the unknot, tied on a finite piece of rope, entangled in such a manner, that they cannot be disentangled to the torus conformation without cutting the rope. The simplest…
Solid experimental evidence has now been obtained that confirms the violation of Bell's inequality in tests of maximally entangled qubit pairs. This violation is widely interpreted as definitive proof of the impossibility of describing…