This book is about “diamond”, a logic of paradox. In diamond, a statement can be true yet false; an “imaginary” state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued Boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown “modulator”; and Metamathematical Dilemma, which relates diamond to Gödelian metamathematics and dilemma games.

More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

The Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal their richness and continued importance for advancing fundamental Physics. If M-Theory is to keep its promise of defining the ultimate structure of matter and spacetime, it is only through the topological configurations of additional dimensionality (or degrees of freedom) that this will be possible. Stretching the exploration of complex space through all of the main equations of Physics should help tighten the noose on “the” fundamental theory. This kind of exploration of higher dimensional spacetime has for the most part been neglected by M-theorists and physicists in general and is taken to its penultimate form here.

1. A sphere -- 2. Surfaces, folds, and cusps -- 3. The inside and outside -- 4. Dimensions -- 5. Immersed surfaces -- 6. Movies -- 7. Movie moves -- 8. Taxonomic summary -- 9. How not to turn the sphere inside-out -- 10. A physical metaphor -- 11. Sarah's thesis -- 12. The eversion -- 13. The double point and fold surfaces

This book is a Festschrift for the 90th birthday of the physicist Pierre Noyes. The book is a representative selection of papers on the topics that have been central to the meetings over the last three decades of ANPA, the Alternative Natural Philosophy Association. ANPA was founded by Pierre Noyes and his colleagues the philosopher-linguist-physicist Frederick Parker-Rhodes, the physicist Ted Bastin, and the mathematicians Clive Kilmister, John Amson.Many of the topics in the book center on the combinatorial hierarchy discovered by the originators of ANPA. Other topics explore geometrical, cosmological and biological aspects of those ideas, and foundational aspects related to discrete physics and emergent quantum mechanics.The book will be useful to readers interested in fundamental physics, and particularly to readers looking for new and important viewpoints in Science that contain the seeds of futurity.

The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph.

1. Classical relativity : scope and beyond. 1.1. Physics and mathematics : long joint journey. 1.2. Inertial motion, relativity, special relativity. 1.3. Space-time as a model of the physical world. 1.4. Generalized theory of relativity and gravitation. 1.5. GRT - first approximation - predictions and tests. 1.6. Exact solutions. 1.7. Observations on the cosmological scale -- 2. Phase space-time as a model of physical reality. 2.1. Preliminary considerations. 2.2. Interpretation dilemma, variation principle, equivalence principle. 2.3. Construction of the formalism. 2.4. Gravitation force in anisotropic geometrodynamics. 2.5. Model of the gravitation source and its applications. 2.6. Electrodynamics in anisotropic space. 2.7. Approaching phase space-time. 2.8. Cosmological picture -- 3. Optic-metrical parametric resonance - to the testing of the anisotropic geometrodynamics. 3.1. Gravitation waves detection and the general idea of opticmetrical parametric resonance. 3.2. OMPR in space maser. 3.3. Astrophysical systems. 3.4. Observations and interpretations. 3.5. On the search for the space-time anisotropy in Milky Way observations

An introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics.

The book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory. Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory. In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V O Manturov and graph-link theory due to both authors. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams. Graph-links can be treated as "diagramless knot theory": such "links" have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.