相关论文: Inscribing smooth knots with regular polygons
An inscribed knot is formed by polygonally connecting points lying on a knot $\gamma$ in parametric order, then closing the path by connecting the first and final points. The stick-knot number of a knot type K is the minimum number of line…
A polynomial knot is a smooth embedding $\kappa: \real \to \real^n$ whose components are polynomials. The case $n = 3$ is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological…
A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is…
We construct a polygonal spiral by arranging a sequence of regular $n$-gons such that each $n$-gon shares a specified side and vertex with the $(n+1)$-gon in the construction. By offering flexibility for determining the size of each $n$-gon…
The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented.
An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges…
We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
In this article it is proven that if a knot, K, bounds an imbedded grope of class n, then the knot is n/2-trivial in the sense of Gusarov and Stanford. That is, all type n/2 invariants vanish on K. We also give a simple way to construct all…
Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…
A generic immersion of a circle into a $2$-sphere is often studied as a projection of a knot; it is called a knot projection. A chord diagram is a configuration of paired points on a circle; traditionally, the two points of each pair are…
A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow \Sigma \times [0,1]$, where $\Sigma$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in…
We use Batson's lower bound on the nonorientable slice genus of $(2n,2n-1)$-torus knots to prove that for any $n \geq 2$, every smooth Jordan curve has an inscribed rectangle of of aspect ratio $\tan(\frac{\pi k}{2n})$ for some $k\in…
In this note, I describe a formalism for treating knots as geometric spaces, and make an application to a simple statistical mechanics computation. The motivation for this study is the natural visual symmetry of the knot, and I describe how…
Given a knot K in the 3-sphere, consider a singular disk bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number…
A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…
Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but…
A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular $n$-gon with edges length $R$. C_2(R) is equivalent to having constant diameter. We show that smooth curves…
A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to…
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these…