Narrow Results

The thickness of a knot is the radius of the thickest rope with which the knot could be tied. Basic properties of thickness have been established. However, thickness is difficult to compute for all but a few knot conformations. Thus, a continuous polygonal thickness function is needed to approximate its smooth analogue. The most natural definition yields incorrect estimates on a planar circle. Here, a polygonal thickness function is defined and shown to be continuous and to correctly approximate smooth thickness with an elementary inscribing algorithm. Examples of thickness estimations are also given.

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C² knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

Let K be a smooth knot of unit thickness embedded in the space with length L(K) and total curvature κ(K). Then where acn(K) is the average crossing number of the embedded knot K and c > 0 is a constant independent of the knot K. This relationship had been conjectured in [G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl.154 (2007) 192–204] where it is shown that the square root power on the curvature is the lowest possible. In the last section we give several examples to illustrate some relationships between the three quantities average crossing number, total curvature and ropelength.

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength-critical configurations for knots and links which are different from the ropelength minima for these knot and link types.

Let Len(K) be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len(K) of a nontrivial knot K in terms of its crossing number c(K) as follows:

The first provably ergodic algorithm for sampling the space of thick equilateral knots off-lattice, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It is an off-lattice generalization of the pivot algorithm. This move to an off-lattice model provides a huge improvement in power and efficacy in that samples can have arbitrary values for parameters such as the thickness constraint, bending angle, and torsion, while the lattice forces these parameters into a small number of specific values. This benefit requires working in a manifold rather than a finite or countable space, which forces the use of more novel methods in Markov–Chain theory. To prove the validity of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint which is used to show that the algorithm is ergodic. This ergodic sampling allows for a statistically valid method for estimating probability distributions of arbitrary functions on the space of thick knots.

For an unoriented link ${𝒦}$, let $L({𝒦})$ be the ropelength of ${𝒦}$. It is known that in general $L({𝒦})$ is at least of the order $O({(\mathrm{\text{Cr}}({𝒦}))}^{3/4})$, and at most of the order $O(\mathrm{\text{Cr}}({𝒦}){ln}^5(\mathrm{\text{Cr}}({𝒦}))$ where $\mathrm{\text{Cr}}({𝒦})$ is the minimum crossing number of ${𝒦}$. Furthermore, it is known that there exist families of (infinitely many) links with the property $L({𝒦})=O(\mathrm{\text{Cr}}({𝒦}))$. A long standing open conjecture states that if ${𝒦}$ is alternating, then $L({𝒦})$ is at least of the order $O(\mathrm{\text{Cr}}({𝒦}))$. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant $a>0$ such that $L({𝒦})\ge aB({𝒦})$ for any ${𝒦}$, where $B({𝒦})$ is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of ${𝒦}$ (called the maximum braid index of ${𝒦}$). Consequently, $L({𝒦})\ge O(\mathrm{\text{Cr}}({𝒦}))$ for any link ${𝒦}$ whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

In this paper, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its convex hull. We measure the convex hull volume of knots during the length annealing process, identifying local minima in the hull volume that arise due to buckling and symmetry breaking. We use $T(p,2)$ torus knots as an illustrative example of a family of knots whose locally minimal-length embeddings are not necessarily ordered by volume. We identify several knots whose central curve has a convex hull volume that is not minimized in the ideal configuration, and find that $8_{19}$ has a non-ideal global minimum in its convex hull volume even when the thickness of its tube is taken into account.