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...

A special type of representation for knots and for local knot manipulations is described and used in a software tool called TOK to implement a number of algorithms on knots. Two algorithms for knot simplification are described: simulated annealing applied to the knot representation, and a “divide-simplify-join” algorithm. Both of these algorithms make use of the compact knot representation and ...

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the ima...

Given a knot presented in a grid diagram, we construct a CW complex which has one cell for each generator of the grid chain complex, and whose homology is the grid homology. This space is welldefined, and its stable homotopy type is a knot invariant. Thus to each knot, we can associate an invariant spectrum, whose F2 homology is the knot Floer homology over F2. 1991 Mathematics Subject Classifi...