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.

In this paper we discuss some fundamental issues regarding knot energy functions. These include the existence of minimum values of energy functions of smooth knots and energy functions of polygonal knots within a knot type, the convergence of these minimum values in the case of polygonal knot energy and the convergence of the corresponding polygons where these minimum values are attained. When the polygonal knot energy is derived from a smooth knot energy, will the minimal polygonal knot energies converge to the infimum of the smooth knot energy? Do the corresponding polygons converge to a smooth knot at which the smooth energy achieves its minimal value? We show that one cannot expect these to be true in general and outline certain conditions that would ensure a positive answer to some of the above questions.

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 knots. The topology of these spaces for the case n=6 and n=7 is described. In both of these cases, each knot space consists of five components,but contains only three (when n=6) or four (when n=7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n≥8 will also be discussed.

We introduce a geometric construction, called a clasp move, which is used to prove several results on stick numbers of links. In particular, we show that all two-component six-crossing links have stick number 8, produce small crossing number knots that can be constructed with fewer sticks than crossings, and improve known upper bounds for the stick number of certain classes of links.

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.

The conformation space of α-regular knots, that is, of spatial polygons which are both equilateral and equiangular, has been investigated by both chemists and mathematicians for quite some time. More recently, the study of α-regular stick numbers, the minimum number of sticks needed to create an α-regular polygon representing a given knot, has also gained some traction. In most of these studies, the value of the angle α has been a fixed constant, playing a secondary role to the number of edges in the polygon or its knot type. In this paper, we consider the entire spectrum of minimal α-regular stick numbers for a knot as a function of α. In particular, we compute this spectrum for the special case of the unknot.

We prove that given any three polygonal unknots in ℝ³, then we may form the Borromean rings out of them through rigid motions of ℝ³ applied to the individual components together with possible scaling of the components. We also prove that if at least two of the unknots are planar, then we do not need scaling. This is true even for a set of three polygonal unknots that are arbitrarily close to three circles, which themselves cannot be used to form the Borromean rings.

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When $n\le 5$, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons.

The probability that a linear embedding of a regular polygon in R³ is knotted should increase as a function of the number of sides. This assertion is investigated by means of an exploration of the compact variety of based oriented linear maps of regular polygons into R³. Asymptotically, an estimation of the probability of knotting is made by means of the HOMFLY polynomial.

We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.

An energy function on knots is a scale-invariant function from knot conformations into non-negative real numbers. The infimum of an energy function is an invariant which defines "canonical conformation(s)" of a knot in three space. These are not necessarily unique, and, in some cases, may even be singular. Many hierarchies of energy functions for knots in the mathematical and physical science literature have been studied, each energy function with its own characteristic set of properties. In this paper we focus on the energy functions of equilateral polygonal knots. These energy functions are important in computer studies of knot energies, and are often defined as discrete versions of energy functions defined on smooth knots. Energy functions on equilateral polygonal knots turn out to be ill-behaved in many cases. To characterize a "good" polygonal knot energy we introduce the concepts of asymptotically finite and asymptotically smooth energy functions of equilateral polygonal knots. Energy functions which are both asymptotically finite and smooth tend to have food limiting behavior (as the number of edges goes to infinity). We introduce a new energy function of equilateral polygonal knots, and show that it is both asymptotically finite and smooth. In addition, we compute this energy for several knots using simulated annealing.

We consider embedding classes of polygonal intervals and circles with fixed edge lengths. We classify all embeddings of polygonal lines of five segments, finding that this space has three connected components for suitable edge lengths. We then construct a class of "stuck" unknotted hexagons, which cannot be deformed to the standard unknotted hexagon. The edge lengths of our examples are not equal.

Polygonal knots are embeddings of polygons in three space. For each n, the collection of embedded n-gons determines a subset of Euclidean space whose structure is the subject of this paper. Which knots can be constructed with a specified number of edges? What is the likelihood that a randomly chosen polygon of n-edges will be a knot of a specific topological type? At what point is a given topological type most likely as a function of the number of edges? Are the various orderings of knot types by means of "physical properties" comparable? These and related questions are discussed and supporting evidence, in many cases derived from Monte Carlo explorations, is provided.

Energy functions on knots are continuous and scale-invariant functions defined from knot conformations into non-negative real numbers. The infimum of an energy function is a knot invariant which defines (not necessarily unique) “canonical conformations” of knots in three space. In this paper we examine strategies of how to find these canonical conformations and compute or measure their minimal energy. Furthermore, we discuss properties that energy functions should have if one wants to compute canonical conformations of knots with minimal energy. Two types of energies are discussed in more detail, the first type consists of energies of C² or C¹ knots defined using the concept of thickness of a knot; the second type is a polygonal energy which should be well suited to numerical computation since it has all the properties discussed earlier.

The probability that a linear embedding of a regular polygon in R³ is knotted should increase as a function of the number of sides . This assertion is investigated by means of an exploration of the compact variety of based oriented linear maps of regular polygons into R³. Asymptotically, an estimation of the probability of knotting is made by means of the HOMFLY polynomial.