Tangle Box

Oh, what a tangled web . . . Everything should have been quiet and pleasant for Ben Holiday, the former Chicago lawyer who became sovereign of the Magic Kingdom of Landover. But it wasn’t. Horris Kew, conjurer, confidence-man, and trickster, had returned to Landover from Ben’s own world. Alas, Horris had not returned of his own volition—he had been sent by the Gorse, a sorcerer of great evil, whom Horris had unwittingly freed from the magic Tangle Box, where it had long ago been imprisoned by the fairy folk. Now it had returned to enslave those who had once dared condemn it. But first, it would rid Landover of all who could stand in its way. . . . Soon Ben found himself imprisoned within the gloom of the Tangle Box, lost in its mists and its labyrinthine ways. The only one who could free Ben from the Tangle Box was the lady Willow. But she had disappeared, was gone from Landover on a mysterious mission of her own. . . .

This invaluable book is an introduction to knot and link invariants as generalized 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 author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this new edition, an article on Virtual Knot Theory and Khovanov Homology has beed added. Contents:Physical KnotsStates and the Bracket PolynomialThe Jones Polynomial and Its GeneralizationsBraids and the Jones PolynomialFormal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)qYang–Baxter Models for Specializations of the Homfly PolynomialKnot-Crystals — Classical Knot Theory in a Modern GuiseThe Kauffman PolynomialThree Manifold Invariants from the Jones PolynomialIntegral Heuristics and Witten's InvariantsThe Chromatic PolynomialThe Potts Model and the Dichromatic PolynomialThe Penrose Theory of Spin NetworksKnots and Strings — Knotted StringsDNA and Quantum Field TheoryKnots in Dynamical Systems — The Lorenz Attractorand selected papers Readership: Physicists and mathematicians. Keywords:Knots;Kauffman;Jones PolynomialReviews: "This book is an essential volume for the student of low-dimensional topology from which a serious student can learn most aspects of modern knot theory. Its informal tone encourages investigation on the part of the reader. The author leaves the reader items to puzzle out." Mathematical Reviews Reviews of the Third Edition: “It is an attractive book for physicists with profuse and often entertaining illustrations … proofs … seldom heavy and nearly always well explained with pictures … succeeds in infusing his own excitement and enthusiasm for these discoveries and their potential implications.” Physics Today “The exposition is clear and well illustrated with many examples. The book can be recommended to everyone interested in the connections between physics and topology of knots.” Mathematics Abstracts “… here is a gold mine where, with care and patience, one should get acquainted with a beautiful subject under the guidance of a most original and imaginative mind.” Mathematical Reviews

This invaluable book is 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 a extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this third edition, a paper by the author entitled "Functional Integration and Vassiliev invariants" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text.

In this thesis, the author develops numerical techniques for tracking and characterising the convoluted nodal lines in three-dimensional space, analysing their geometry on the small scale, as well as their global fractality and topological complexity---including knotting---on the large scale. The work is highly visual, and illustrated with many beautiful diagrams revealing this unanticipated aspect of the physics of waves. Linear superpositions of waves create interference patterns, which means in some places they strengthen one another, while in others they completely cancel each other out. This latter phenomenon occurs on 'vortex lines' in three dimensions. In general wave superpositions modelling e.g. chaotic cavity modes, these vortex lines form dense tangles that have never been visualised on the large scale before, and cannot be analysed mathematically by any known techniques.

The motivations, goals and general culture of theoretical physics and mathematics are different. Most practitioners of either discipline have no necessity for most of the time to keep abreast of the latest developments in the other. However on occasion newly developed mathematical concepts become relevant in theoretical physics and the less rigorous theoretical physics framework may prove valuable in understanding and suggesting new theorems and approaches in pure mathematics. Such interdis ciplinary successes invariably cause much rejoicing, as over a prodigal son returned. In recent years the framework provided by quantum field theory and functional in tegrals, developed over half a century in theoretical physics, have proved a fertile soil for developments in low dimensional topology and especially knot theory. Given this background it was particularly pleasing that NATO was able to generously sup port an Advanced Research Workshop to be held in Cambridge, England from 6th to 12th September 1992 with the title Low Dimensional Topology and Quantum Field Theory. Although independently organised this overlapped as far as some speak ers were concerned with a longer term programme with the same title organised by Professor M Green, Professor E Corrigan and Dr R Lickorish. The contents of this proceedings of the workshop demonstrate the breadth of topics now of interest on the interface between theoretical physics and mathematics as well as the sophistication of the mathematical tools required in current theoretical physics.

This volume includes both rigorous asymptotic results on the inevitability of random knotting and linking, and Monte Carlo simulations of knot probability at small lengths. The statistical mechanics and topology of surfaces on the d-dimensional simple cubic lattice are investigated. The energy of knots is studied both analytically and numerically. Vassiliev invariants are investigated and used in random knot simulations. A mutation scheme which leaves the Jones polynomial unaltered is described. Applications include the investigation of RNA secondary structure using Vassiliev invariants, and the direct experimental measurement of DNA knot probability as a function of salt concentration in random cyclization experiments on linear DNA molecules. The papers in this volume reflect the diversity of interest across science and mathematics in this subject, from topology to statistical mechanics to theoretical chemistry to wet-lab molecular biology.

An acorn falls far from the tree Former Chicago lawyer Ben Holiday was very proud and quite happy. And why shouldn’t he be? The Magic Kingdom which he ruled as High Lord was finally at peace. He was free to lie back and watch as his new daughter grew. And grow she did—by leaps and bournds, shooting through infancy in a matter of months. She took her first steps and learned to swim in the same week. An amalgam of magic and heredity, Mistaya was born a seedling, nourished by soils from Landover, Earth, and the fairy mists, come into being in the dank, misty deadness of the Deep Fell. She was as lovely as her mother, the sylph Willow, with dazzling green eyes that cut to the soul. Ben wished he could enjoy his daughter's childhood and his happy kingdom forever. Alas, those idyllic days were not to last. For Rydall, king of lands beyond the fairy mist, rode up to the gates of Sterling Silver and shattered peace of Landover. His armies were poised on the border, ready to invade unless Ben accepted a challenge: Rydall would send seven champions to face Ben, each in a different form. If Ben triumphed over all seven, Rydall would then abandon his claims to the kingdom. Some counseled the High Lord to refuse Rydall's challenge, but Holiday could not, for Mistaya had been snatched from her guardians by foul magic. And Rydall held the key to her fate. . . .

After New York Times bestselling author Terry Brooks completed The Sword of Shannara trilogy, millions of fans around the world clamored to immerse themselves again in his dazzling world. Brooks answered with a quartet of beloved novels—The Heritage of Shannara. Now, for the first time in one handsome collector’s edition hardcover, here are: The Scions of Shannara, The Druid of Shannara, The Elf Queen of Shannara, and The Talismans of Shannara—the thrilling continuation of the saga that has become one of the most enduring fantasy epics of all time. The Scions of Shannara Since the death of the Druid Allanon, magic has been strictly forbidden in the Four Lands. Yet Par still has limited use of the Wishsong. Then a dire message from Allanon sends Par and his brother Coll on an impossible task: to recover the long-lost Sword of Shannara—or all life in the Four Lands will be destroyed. The Druid of Shannara Evil forces remain in control of the Four Lands. To restore the Keep of the Druid Allanon, Walker Boh sets out on a journey to find the black Elfstone. He must venture into perilous, unknown lands with a strange band of fellow travelers—one of whom is hatching his own sinister plot. The Elf Queen of Shannara Ordered by the Druid Allanon to find the Elves and return them to the world of Men, Wren is carried away to an island where Elves might still exist. Even if by some miracle she locates the Elves, can she convince them to follow her back through a demon-haunted jungle to the safety of the shore? The Talismans of Shannara The Shadowen still swarm over the Four Lands, poisoning everything with their dark magic. Their leader is determined to destroy all the Scions of Shannara. With traps cleverly laid, the charges are doomed to failure—unless Par can discover a way to harness the power of the Sword of Shannara. So continues Terry Brooks’s enthralling Heritage of Shannara epic, a spellbinding saga that chronicles the ultimate battle of good vs. evil. BONUS: This edition contains an excerpt from Terry Brooks's The Measure of the Magic.