A Lemma on Systems of Knotted Curves

Abstract

continuous correspondences on a Riemann surface, whether algebraic or not, uithout recourse to transcendental considerations.(d) Open manifolds. Here an adaptation of a reasoning due to Alexander leads to the solution of the question.(e) One-sided manifolds. They may be replaced by suitable two-sided open manifolds. (f) Conformal representation. Julia's theorem on the conformal representation of a plane region on a part of itself,3 together with Ritt's extension4 and other generalizations for functions of several complex variables can be readily obtained by our method.Consider a system S made up of a finite number of simple noninteresting closed curves located in real euclidean 3 space. The curves S may be arbitrarily knotted and linking, but we shall assume, in order to simplify matters as much as possible, that each is composed of a finite number of straight pieces. The problem will be to prove that the system S is always topologically equivalent (in the sense of isotopic) to a simpler system S', where S' is so related to some fixed axis in space that as a point P describes a curve of S' in a given direction the plane through the axis and the point P never ceases to rotate in the same direction about the axis. An application of this lemma to the theory of 3-dimensional manifolds will be given at the end of the communication.It will be convenient to visualize the system S by means of its projection S, upon a plane. By choosing the center of projection in general position, the projection S,. will have no other singularities than isolated double points at each of which a pair of straight pieces actually cross one another. Wherever a double point occurs, it will be necessary to indicate which of the two branches is to be thought of as passing behind the other, either by removing a little segment from the branch in question or by some equivalent device. The problem will then be to transform the figure S,, by 93 VOiL. 9, 1923

Open book decompositions of fiber sums in contact topology

“…According to a theorem of Alexander [1] every closed oriented 3-manifold admits a so-called open book decomposition. While it had been known for almost 40 years that open books carry a natural contact structure [11], at the beginning of the millennium it turned out that this was just one fragment of a much deeper correlation.…”

On cobordisms between knots, braid index, and the Upsilon-invariant

“…A classical theorem of Alexander states that every knot arises as the closure of an n-braid for some positive integer n [Ale23]. Here an n-braid is an element of Artin's braid group on n-strands B n [Art25].…”

Envisioning Transformations—The Practice of Topology

“…Actual pictures trigger our imagination and help us see modifications on them, but for the people who are already acquainted with a practice on pictures of a certain type (e.g., links or braids) it is perhaps not necessary any more to actually draw all the pictures. As previously mentioned, the original proof by Alexander did not contain any single figure (Alexander 1923). For the experts, what matters is the spatial configurations that are displayed by the figures and not their appearances.…”

“…We introduce now Alexander's theorem and give a proof that follows the original one, which can be found in (Alexander 1923). In order to make the proof more A video would be very effective to show this isotopy.…”