Möbius Transformations of Polygons and Partitions of 3-SPACE (original) (raw)

Complementary Regions of Knot and Link Diagrams

Annals of Combinatorics, 2011

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are all universal for knots and links: (3, 5, 7,. . .), (2, n, n + 1, n + 2,. . .) for all n ≥ 3 and (3, n, n + 1, n + 2,. . .) for all n ≥ 4. Moreover, the following finite sequences are also universal for knots and links: (3, 4, 5) and (2, 4, 5). It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.

Monte Carlo explorations of polygonal knot spaces

Knots in Hellas, 2000

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.

Knotted polygons with curvature in

Journal of Physics A: Mathematical and General, 1998

The knot probability of semiflexible polygons on the cubic lattice is investigated. The degree of stiffness of the polygon is mimicked by introducing a bending fugacity conjugate to the curvature of the polygon. By generalizing Kesten's pattern theorem to semiflexible walks, we show that for any finite value of the bending fugacity all except exponentially few sufficiently long polygons are knotted.

Obtaining graph knots by twisting unknots

Topology and its Applications, 2005

Let K be a knot in the 3-sphere S 3 , and D a disk in S 3 meeting K transversely more than once in the interior. For non-triviality we assume that |D ∩ K| ≥ 2 over all isotopies of K in S 3 − ∂D. Let K D,n (⊂ S 3) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and K D,n is a graph knot, then |n| ≤ 1 or K and D form a special pair which we call an "exceptional pair". As a corollary, if (K, D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.

Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ³

Contemporary Mathematics, 2002

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An Investigation of Polygonal Knot Spaces and Ideal Physical Knot Configurations

Spaces of polygonal knots, subject to specified constraints such as the number of nondegenerate edges or the requirement of having fixed edge lengths, provide the context within which it is appropriate to study configurations which are ideal with respect to a variety of natural physically motivated constraints. Even for polygonal knots with relatively few vertices, the high dimensionality and complexity of the knot space structure makes analytical investigations impractical. In this note we will discuss the methods and the results of a Monte Carlo investigation of several fundamental approaches to ideal polygonal knot configurations for small numbers of edges.

Knots Obtained by Twisting Unknots

Let K be the unknot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior, at least twice (after all isotopies). We denote by KD,n a knot obtained from K by n twistings along the disk D. We describe for which pairs (K,D) and integers n, KD,n is a torus knot, a satellite knot or a hyperbolic knot.