Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knots. From this fine structure and an analysis of the associated planar graph, one can define a measure of knot complexity using the number of independent unknotting pathways from the global knot type to the short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on ...

We consider the relations ≥ and ≥p on the collection of all knots, where k ≥ k (respectively, k ≥p k) if there exists an epimorphism πk → πk of knot groups (respectively, preserving peripheral systems). When k is a torus knot, the relations coincide and k must also be a torus knot; we determine the knots k that can occur. If k is a 2-bridge knot and k ≥p k, then k is a 2-bridge knot with determ...

This work presents a method for associating a class of constraint satisfaction problems to a three-dimensional knot. Given a knot, one can build a knot quandle, which is a generally an infinite, free algebra. The desired collection of problems is derived from the finite quotients of the knot quandle by applying theory that relates finite algebras to constraint languages. Along the way, notions ...

Almost all energies are not homothety invariant, so we will consider only knots of length 2π. The energy of a knot is not an invariant of the topological class of this knot. If we make a smooth perturbation of a knot, its energy smoothly changes. We will consider energies with the following important properties. The energy is always positive. When a knot crossing tends to a double point, the en...

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

Introduction With arthroscopic techniques being used, the importance of knot tying has been examined. Previous literature has examined the use of reversing half-hitches on alternating posts (RHAPs) on knot security. Separately, there has been research regarding different suture materials commonly used in the operating room. The specific aim of this study was to validate the effect of different ...

Let K be a knot in the 3-sphere S3, and D a disc in S3 meeting K transversely more than once in the interior. For non-triviality we assume that |K ∩D| ≥ 2 over all isotopy of K. Let Kn(⊂ S3) be a knot obtained from K by cutting and n-twisting along the disc D (or equivalently, performing 1/n-Dehn surgery on ∂D). Then we prove the following: (1) IfK is a trivial knot andKn is a composite knot, t...

An alternating diagram encodes a lot of information about a knot. For example, if an alternating knot is composite, this is evident from the diagram [10]. Also, its genus ([3], [12]) and its crossing number ([7], [13], [17]) can be read off directly. In this paper, we apply this principle to alternating knots with tunnel number one. Recall that a knot K has tunnel number one if it has an unknot...

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the “boundary knot” that occurs around the singular set, the set of points at which the embedding is not lo...

Fifty umbilical cords were prepared and examined and their venous perfusion pressures measured with and without a true knot in the cord. Contrary to information in the literature, a loose umbilical cord knot did not affect the venous perfusion pressure. With a tightened knot, the smaller the umbilical cord diameter, the greater was the pressure required to perfuse past the knot. The umbilical v...