Article

Old and New Results on Knots

January 1950
Canadian Journal of Mathematics 2

DOI:10.4153/CJM-1950-001-7

Authors:

The theory of knots undertakes the task of giving a complete survey of all existing knots. A solid mathematical foundation was not laid to this theory until our century. A mathematician of the rank of Felix Klein thought it to be nearly hopeless to treat knot problems with the same exactness as we are accustomed to from classical mathematics. We want to give here a short summary of the modern topological methods enabling us to approach the knot problem in a mathematical way. In order to exclude pathological knots, as for instance knots being entangled an infinite number of times, we will define a knot as a polygon lying in the space. In other words: a knot is a closed sequence of segments without double points. In Figure 1 some examples of knots are given in plane projection.

The theory of one-relator groups: history and recent progress

Preprint

Full-text available

Jan 2025

The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.

Mariposas y 3-variedades

Article

Full-text available

Mar 2004

En este artículo mostraremos que para cada diagrama de un nudo 3-coloreable se le puede asociar, en forma canónica, una 3-bola (llamada mariposa) con caras identificadas por reflexiones topológicas, la cual se puede triangular y 4-colorear, de tal manera que la coloración del nudo y de la triangulación resultan compatibles. Como corolario se obtiene que para toda 3-variedad hay al menos una bola asociada a ella con dichas características. Abstract In this paper we show that for any 3-colorable knot diagram there is a canonic 3-ball, called butterfly, with faces that are identified by topological reflections. Moreover, we prove that this ball admits a 4-colored triangulation compatible with the 3-coloration of the knot. As a corollary we associate to any 3-manifold at least one of these balls.

Vier Heidelberger Topologen 1935–1996

Article

Full-text available

May 2022

Klaus Volkert

We present the lives and the work of four topologists who contributed to establish a center of topology at Heidelberg: Herbert Seifert, William Threlfall, Albrecht Dold and Dieter Puppe.

Topology of Magnetic Fields in Particle Physics, Implications on the Quark Model

Chapter

Jan 1977

Herbert Jehle

On double-torus knots. II

Article

Aug 2000

A double-torus knot is a knot embedded in a genus two Heegaard surface T in S3. We consider double-torus knots L such that T - L is connected, and consider fibred knots in various classes.

On the Alexander Polynomial

Article

Jan 1953
ANN MATH

Guillermo Torres

On Double-Torus Knots

Article

Dec 1999

Peter Clifford Hi

A double-torus knot is a knot embedded in a genus two Heegaard surface T in S-3. After giving a notation for these knots, we consider double-torus knots L such that T - L is not connected, and give a criterion for such knots to be non-trivial. Various new types of non-trivial knots with trivial Alexander polynomial are found.

Linking Spheres

Article

Apr 1960

E. Zeeman

Electron-muon puzzle and the electromagnetic coupling constant

Article

Jun 1977

Herbert Jehle

On the basis of a heuristic model we argued in an earlier paper (paper C of this series) electric field (and of course the magnetic field, too) of a lepton or of a quark may be formulated in terms of a closed loop of quantized magnetic flux whose alternative forms ("loopforms") are superposed with probability amplitudes so as to represent the electromagnetic field of that lepton or quark. The Zitterbewegung of a single stationary ("elementary") particle suggests a kind of quasiextension, which is assumed, in the present theory, to permit concepts of structuralization of the electromagnetic field even for leptons. Mesons and baryons may be represented by linked quantized flux loops, i.e., quark loops (as in paper B). The central problem now (in this paper D) is to formulate those probability-amplitude distributions in terms of wave functions to characterize the internal structure of the lepton or quark in question. As probability-amplitude functions one may choose bases of irreducible representations of the group with respect to which the model is to be invariant. It is seen that this implies the SO(4) group. As both the electron-muon mass ratio and the electromagnetic coupling constant depend, in this flux-quantization model, on the correct formulation of the structuralization of probability-amplitude distributions, we should expect to get an insight into both these puzzles from finding the right probability-amplitude wave functions. Furthermore, it is seen that this same structuralization of probability-amplitude distributions also permits one to estimate the rate of weak interactions, thus relating them to electromagnetic interactions.

Flux quantization and fractional charges of quarks

Article

Apr 1975

Herbert Jehle

Two distinct objectives, (A) and (B), characterize this project of flux quantization and particle physics. (A) proposes that, instead of starting with electric point charges to derive magnetic and other properties of (elementary) particles, one may consider spinning, closed loops of quantized flux Φq=hc/e as the elementary constituents ("elementary loops") from which electric and other properties of particles are derived. The manifold of alternative forms of one single loop (which follow "fibrations" of ordinary space) represents a lepton. In terms of a heuristic model, the consistency of this program had been shown, and shown to imply the derivation of the electromagnetic coupling constant e2/ℏc. (B) relates the classification of particles and their conservation laws to the topology of flux loops and of their interlinkage. A magnetic field formed from nonintersecting "loopforms" implies topologically the forms of torus knots. The toroidal fibrations of ordinary three-space by two and three coaxially interlinked quarkloops represent mesons and baryons. The loops may independently spin about their common central symmetry axis and "whirl" about their common circular torus midline; effective spin (versus flux) orientation determines their equivalent electric charges. N, P, and λ quarks correspond to fibrations of space in terms of loops of "winding numbers" (2, 1), (3, 1), and (3, 2), respectively; loops (3, ±2) have "unknotting numbers" ± 1 corresponding to strangeness ± 1. Electromagnetic interactions are, in the conventional way, understood as interactions of distant loops through their electromagnetic field, strong interactions as merging or creation of loop-antiloop pairs, as well as the various different types of exchanges of quark loops between the interacting particles, and weak interactions as crossing of loops over themselves or over those they interact with. This implies that strangeness-violating interactions are weak and parity-violating. The present objective (C) is to consider the existence of the two different types of spin-whirl motion, one of which has the same handedness as that of the fibration, in which case the effective motion of loopforms is subtractively composed of spin and whirl; such was assumed in (B) to characterize quarks. In the other type, i.e., opposite handedness of fibration and of spin-whirl motions, these motions contribute additively to effective motion of torus knots; loops of winding numbers (2, 1) then characterize electrons and muons. The riddle of fractional charges of quarks disappears in this theory. So, also, does the riddle of quark statistics (symmetric spin-isospin functions of baryons) because their linkage makes quark loops localized objects. The flux loops may properly be called "elementary loops." The next objective (D) concerns the probability amplitudes which characterize the distribution of the forms of a torus-knot flux loop. They are shown to be specified by an SO(4)-invariant formulation. To know which group representations to choose from is a prerequisite to a completion of objective (A), and to a quantitative specification of (B).

Gauge Conditions in the Singular Systems

Article

Aug 2008

In terms of the Dirac's finite contact transformation, we derive the principle on the choice of gauge conditions in the general singular Lagrangian system. It is applied successfully to Cawley's first counter-example of Dirac's conjecture. The number of gauge conditions or gauge freedoms is given and it is shown to be different from the accustomed conclusion which is always equal to the number of all the first-class constraints.

Topological geometrodynamics

Article

Full-text available

Sep 1986

M. Pitkanen

The description of 3-space as a spacelike 3-surfaceX of the spaceH = M 4 ×CP 2 (Product of Minkowski space and two-dimensional complex projective spaceCP 2) and the idea that particles correspond to 3-surfaces of finite size inH are the basic ingredients of topological geometrodynamics (TGD), an attempt at a geometry-based unification of the fundamental interactions. The observations that the Schrödinger equation can be derived from a variational principle and that the existence of a unitaryS-matrix follows from the phase symmetry of this action lead to the idea that quantum TGD should be derivable from a quadratic phase-symmetric variational principle for some kind of superfield (describing both fermions and bosons) in the configuration space consisting of the spacelike 3-surfaces ofH. This idea as such has not led to a calculable theory. The reason is the wrong realization of the general coordinate invariance. The crucial observation is that the space Map(X, H), the space of maps from an abstract 3-manifoldX toH, inherits a coset space structure fromH and can be given a Kahler geometry invariant under the local M4×SU(3) and under the group Diff ofX diffeomorphisms. The space Map(X, H) is taken as a basic geometric object and general coordinate invariance is realized by requiring that superfields defined in Map(X, H) are diffeo-invariant, so that they can be regarded as fields in Map(X, H)/Diff, the space of surfaces with given manifold topology. Superd'Alembert equations are found to reduce to a simple algebraic condition due to the constant curvature and Kähler properties of Map(X, H). The construction of physical states leads by localM 4× SU(3) invariance to a formalism closely resembling the quantization of strings. The pointlike limit of the theory is discussed. Finally, a formal expression for theS-matrix of the theory is derived and general properties of theS-matrix are discussed.

A survey of semigroups of continuous selfmaps

Article