Inscribing smooth knots with regular polygons

This is an expository article on diagrammatic representations of knots and links in various settings via braids.

In this short article I introduce the knotR package, which creates two dimensional knot diagrams optimized for visual appearance using the R programming language. The knotR package is a systematic R-centric suite of software for the creation of production-quality artwork of knot diagrams, released under GPL2.

We explore under what conditions one can obtain a nontrivial knot, given a collection of $n$ vectors. First, we show how to get a crossing from any 3 vectors equal in magnitude, by arbitrarily picking 2 vectors and identifying the sufficient and necessary criteria for picking a third vector that will guarantee a crossing when the vectors are reordered. We also show that it's always possible for a set of vectors to be reordered to form the unknot, if they sum to $\vec{0}$ when...

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $L/k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular $n$-gon admits a $1/2n$-geodesic. For the doubled regular $p$-gons, with $p$ an odd prime, we conjecture that $k=2p$ is the minimum value for $k$ such that the space admits a $1/k$-geodesic.

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q+1,q) torus knot is (2q+1)cot(\pi/(2q+1)) (resp., 2q cot(\pi/(2q+1))). Using these calculations, we provide t...

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are $C^{1,1}$ curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.

For every odd integer $N$ we give an explicit construction of a polynomial curve $\cC(t) = (x(t), y (t))$, where $\deg x = 3$, $\deg y = N + 1 + 2\pent N4$ that has exactly $N$ crossing points $\cC(t_i)= \cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pad\'e approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergence, with that discovery aided by computational experiments. Sufficient conditions to attain isotopic equivalence can be determined a priori. These sufficient con...

This paper visualizes a knot reduction algorithm

In this paper we show how to realize all knot (and link) types as C^{2} smooth curves of constant curvature. Our proof is constructive: we build the knots with copies of a fixed finite number of "building blocks" that are particular segments of helices and circles. We use these building blocks to construct all closed braids.