Inscribing smooth knots with regular polygons

We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ \deg C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 4.$

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the trajectories of energy descent upon the initial geometric conditions of the knotted embedding.

The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.

We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.

I briefly discuss a method of obtaining distinct classes of topologically equivalent knots by developing appropriate computer programs.

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 segments needed to polygonally form some knot with the same homotopy type. The stick-knot number of a trefoil is 6. Cole Hugelmeyer studied the manifold $M$ consisting of 6 points lying on a triangular prism and found that by intersecting a...

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 this might be carried out. The direct approach, however, fails due to limits of the visual symmetry, but by recasting the problem in terms of the geometry of contours of the knot, the resulting permutation operators provide a better analyt...

We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a two com...

We show that every proper, smooth 2-knot is ambient isotopic to a polynomial embedding from $\mathbb{R}^2$ to $\mathbb{R}^4$. This representation is unique up to a polynomial isotopy. Using polynomial representation of classical long knots we show that all twist spun knots posses polynomial parametrization. We construct such parametrizations for few spun and twist spun knots and provide their $3$ dimensional projections using Mathematica.

We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on placing knots on a cubic lattice.