Knots and Applications

The following sections are included:

• Preface

• References

• Table of Contents

Knot and link diagrams are used to represent nonstandard sets, and to represent the formalism of combinatory logic (lambda calculus). These diagrammatics create a two-way street between the topology of knots and links in three dimensional space and key considerations in the foundations of mathematics.

The following sections are included:

• ON VORTEX ATOMS

• ON VORTEX MOTION

• THE TRANSLATORY VELOCITY OF A CIRCULAR VORTEX RING

• VORTEX STATICS

Sufficiently nonlinear classical fields admit modes called kinks, whose number is strictly conserved in virtue of boundary conditions and continuity of the field as a function of space and time. In a quantum theory of such fields, with canonical commutation (not anticommutation) relations, kinks and their conservation still persist, and even if the intrinsic angular momentum is an integer, a rotating kink can have half-odd angular momentum, if double-valued state functionals are admitted. We formulate a natural concept of exchange appropriate for kinks. The principal result is that for fields with integer-valued intrinsic angular momentum, the observed relation between spin and (exchange) statistics follows from continuity alone, parastatistics being excluded. It is likely that in the theories with even (odd) exchange statistics, suitable creation operators will commute (anticommute). We show that, while the rotational spectrum of a kink will in general possess both integer and half-odd spin states, in fields with integer-valued intrinsic angular momentum only one of these two possibilities will ever be observed for each kind of kink, and that there is a nonzero “particle number” (strictly conserved, additive, scalar quantum number) attached to half-odd-spin kinks of each kind. It then follows that a boson and a fermion kink will always differ in at least one particle number, as well as in spin, and that, in particular, every fermion kink will have some nonzero particle number. These results are consistent with the hypothesis that the spinor fields usually employed to describe half-odd-spin quanta are not fundamental, but are useful “point-limit” approximations to operators creating or annihilating excitations in a nonlinear field of particular kinds of kinks in particular internal states.

Quantized flux has provided an interesting model for muons and for electrons: One closed flux loop of the form of a magnetic dipole field line is assumed to adopt alternative forms which are superposed with complex probability amplitudes to define the magnetic field of a source lepton. The spinning of that loop with an angular velocity equal to the Zitterbewegung frequency 2mc²/ħ implies an electric Coulomb field, (negative) positive, depending on (anti) parallelism of magnetic moment and spin. The model implies CP invariance. A quark may be represented by a quantized flux loop if interlinked with another loop in the case of a meson, with two other loops in the case of a baryon. Because of the link, their spinning is very different from that of a single loop (lepton). The concept of a single quark does not exist accordingly, and it is seen that a baryon with a symmetric spin-isospin function in the SU(2) × SU(3) quark representation might not violate the Pauli principle because the wave function representing the relative position of linked loops may be chosen antisymmetric. Weak interactions may be understood to occur when the flux loops involved in the interaction have to cross over themselves or over each other. Strangeness is readily interpreted in terms of the trefoil character of a λ quark: Strangeness-violating interactions imply crossing of flux lines and are thus weak and parity-nonconserving. ΔS = ΔQ is favored in such interactions. Intrinsic symmetries may be interpreted in terms of topology of linked loops. Sections I and II give a short résumé of the 1971 paper.

The familiar wormhole model of geometrodynamics is extended to allow for knotted embeddings of the initial hypersurface. It is shown that topology change is not only a means to modify the connectivity of the space, but also the knot invariants of its embedding. In a probabilistic framework the process of “wormhole scattering” can be expressed by creation and annihilation operators acting on the wave function of quantum geometrodynamics. Implications concerning Wheeler's exciton model of elementary particles, the f-gravity approach to hadronic matter, and interrelations with Jehle's flux quantization program are discussed.

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘ intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

We review two different methods of calculating Witten's invariant: a stationary phase approximation and a surgery calculus. We give a detailed description of the 1-loop approximation formula for Witten's invariant and of the technics involved in deriving its exact value through a surgery construction of a manifold. Finally we compare the formulas produced by both methods for a 3-dimensional sphere S³ and a lens space L(p, 1).

A substantial understanding of quantum theory is not a prerequisite for reading this paper.

We show that 2+1-dimensional Euclidean quantum gravity is equivalent, under some mild topological assumptions, to a Gaussian fermionic system. Furthermore we argue that the corresponding 2+1-dimensional Euclidean quantum gravity partition function may be related to the partition function of the reduced 3D-lattice Ising model for finite lattices.

The following sections are included:

• Introduction

• Foldings and the Brauer Monoid

• Embedded Foldings, Graph Invariants and the Vassiliev Invariants

• Generalizing The Invariant I

• More about Vassiliev invariants

• Discussion

• Appendix 1

• Appendix 2: Complexity and Proximity of Foldings

• REFERENCES

The following sections are included:

• Introduction

• Background of Polymer Molecular Entanglement

• Introduction to the Topology of Entanglement

• What is a Polymer

• Analyzing Tangled Systems

• Selection of Model Structures

• Entanglement Models

• Entanglement by Interlocking Loops

• An Unhindered System of Mobile Loops

• Application to Systems with Selective Immobilization

• References

Topology has suggested several classic targets for chemical synthesis, including molecular Möbius strips. Details of the first synthesis of a molecular Möbius strip, possessing a three rung Möbius ladder molecular graph, are described. The structure is composed of crown ether rings fused by the tetrahydroxymethylethylene (THYME) unit—the rungs of the Möbius ladder are thus carbon-carbon double bonds and the single edge is a polyether chain of 60 atoms. A chemical realization of the classic topological trick of cutting a Möbius strip “in half” only to have it remain in one piece is accomplished by ozonolysis of the double bonds, and serves as an elegant corroboration of the topology of the structure. In addition, just as topology suggested targets for synthesis, the synthesis has stimulated new topology. A brief description of some of these new mathematical results is presented, including the discovery of a new and very “deep” kind of chirality. Thus the three rung THYME Möbius ladder molecular graph serves as the prototype of the class of intrinsically chiral graphs—possessing a kind of chirality which can never occur in knots or linked rings.

Why do impossible figures, which cannot exist in three dimensions, appear to make three-dimensional sense? In order to shed some light on this question the limits may be tested to which three-dimensional operations on these figures can be performed. In this paper a particularly difficult operation, viz., torus eversion is attempted. Not only is an eversion found to be possible but an unfamiliar impossibility develops. The regular form of the eversion is shown to be unique.

The following sections are included:

• List of Authors