Polynomial One Cocycles For Knots And Closed Braids

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Polynomial One-cocycles For Knots And Closed Braids

Fiedler Thomas
Polynomial One cocycles For Knots And Closed Braids Book PDF

Author : Fiedler Thomas
Publisher : World Scientific
Page : 260 pages
File Size : 54,9 Mb
Release : 2019-08-27
Category : Mathematics
ISBN : 9789811210310

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Polynomial One-cocycles For Knots And Closed Braids by Fiedler Thomas Pdf

Traditionally, knot theory deals with diagrams of knots and the search of invariants of diagrams which are invariant under the well known Reidemeister moves. This book goes one step beyond: it gives a method to construct invariants for one parameter famillies of diagrams and which are invariant under 'higher' Reidemeister moves. Luckily, knots in 3-space, often called classical knots, can be transformed into knots in the solid torus without loss of information. It turns out that knots in the solid torus have a particular rich topological moduli space. It contains many 'canonical' loops to which the invariants for one parameter families can be applied, in order to get a new sort of invariants for classical knots.

One-cocycles And Knot Invariants

Thomas Fiedler
One cocycles And Knot Invariants Book PDF

Author : Thomas Fiedler
Publisher : World Scientific
Page : 341 pages
File Size : 44,5 Mb
Release : 2023-01-04
Category : Mathematics
ISBN : 9789811263019

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One-cocycles And Knot Invariants by Thomas Fiedler Pdf

One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.

Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science - Volume 1: The Golden Section, Fibonacci Numbers, Pascal Triangle, And Platonic Solids

Alexey Stakhov
Mathematics Of Harmony As A New Interdisciplinary Direction And Golden Paradigm Of Modern Science Volume 1 The Golden Section Fibonacci Numbers Pascal Triangle And Platonic Solids Book PDF

Author : Alexey Stakhov
Publisher : World Scientific
Page : 247 pages
File Size : 44,7 Mb
Release : 2020-05-05
Category : Mathematics
ISBN : 9789811206382

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Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science - Volume 1: The Golden Section, Fibonacci Numbers, Pascal Triangle, And Platonic Solids by Alexey Stakhov Pdf

Volume I is the first part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science-volume 3:the "Golden" Paradigm Of Modern Science: Prerequisite For The "Golden" Revolution In Mathematics,computer Science,and Theoretical Natural Sciences

Alexey Stakhov
Mathematics Of Harmony As A New Interdisciplinary Direction And Golden Paradigm Of Modern Science volume 3 the Golden Paradigm Of Modern Science Prerequisite For The Golden Revolution In Mathematics computer Science and Theoretical Natural Sciences Book PDF

Author : Alexey Stakhov
Publisher : World Scientific
Page : 244 pages
File Size : 42,5 Mb
Release : 2020-09-03
Category : Mathematics
ISBN : 9789811213519

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Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science-volume 3:the "Golden" Paradigm Of Modern Science: Prerequisite For The "Golden" Revolution In Mathematics,computer Science,and Theoretical Natural Sciences by Alexey Stakhov Pdf

Volume III is the third part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

On Complementarity: A Universal Organizing Principle

Jack Shulman Avrin
On Complementarity A Universal Organizing Principle Book PDF

Author : Jack Shulman Avrin
Publisher : World Scientific
Page : 281 pages
File Size : 55,8 Mb
Release : 2021-08-06
Category : Science
ISBN : 9789813278998

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On Complementarity: A Universal Organizing Principle by Jack Shulman Avrin Pdf

It is not uncommon for the Principle of Complementarity to be invoked in either Science or Philosophy, viz. the ancient oriental philosophy of Yin and Yang whose symbolic representation is portrayed on the cover of the book. Or Niels Bohr's use of it as the basis for the so-called Copenhagen interpretation of Quantum Mechanics. This book arose as an outgrowth of the author's previous book entitled 'Knots, Braids and Moebius Strips,' published by World Scientific in 2015, wherein the Principle itself was discovered to be expressible as a simple 2x2 matrix that summarizes the algebraic essence of both the well-known Microbiology of DNA and the author's version of the elementary particles of physics. At that point, the possibility of an even wider utilization of that expression of Complementarity arose.The current book, features Complementarity, in which the matrix algebra is extended to characterize not only DNA itself but the well-known process of its replication, a most gratifying outcome. The book then goes on to explore Complementarity, with and without its matrix expression, as it occurs, not only in much of physics but in its extension to cosmology as well.

Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science - Volume 2: Algorithmic Measurement Theory, Fibonacci And Golden Arithmetic's And Ternary Mirror-symmetrical Arithmetic

Alexey Stakhov
Mathematics Of Harmony As A New Interdisciplinary Direction And Golden Paradigm Of Modern Science Volume 2 Algorithmic Measurement Theory Fibonacci And Golden Arithmetic s And Ternary Mirror symmetrical Arithmetic Book PDF

Author : Alexey Stakhov
Publisher : World Scientific
Page : 331 pages
File Size : 47,9 Mb
Release : 2020-09-03
Category : Mathematics
ISBN : 9789811213489

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Mathematics Of Harmony As A New Interdisciplinary Direction And "Golden" Paradigm Of Modern Science - Volume 2: Algorithmic Measurement Theory, Fibonacci And Golden Arithmetic's And Ternary Mirror-symmetrical Arithmetic by Alexey Stakhov Pdf

Volume II is the second part of the 3-volume book Mathematics of Harmony as a New Interdisciplinary Direction and 'Golden' Paradigm of Modern Science. 'Mathematics of Harmony' rises in its origin to the 'harmonic ideas' of Pythagoras, Plato and Euclid, this 3-volume book aims to promote more deep understanding of ancient conception of the 'Universe Harmony,' the main conception of ancient Greek science, and implementation of this conception to modern science and education.This 3-volume book is a result of the authors' research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the 'Mathematics of Harmony,' a new interdisciplinary direction of modern science. This direction has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the generalized Binet's formulas), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational bases, Fibonacci computers, ternary mirror-symmetrical arithmetic).The books are intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science.

Scientific Legacy Of Professor Zbigniew Oziewicz: Selected Papers From The International Conference "Applied Category Theory Graph-operad-logic"

Hilda Maria Colin Garcia,Jose De Jesus Cruz Guzman,Louis H Kauffman,Hanna Makaruk
Scientific Legacy Of Professor Zbigniew Oziewicz Selected Papers From The International Conference Applied Category Theory Graph operad logic Book PDF

Author : Hilda Maria Colin Garcia,Jose De Jesus Cruz Guzman,Louis H Kauffman,Hanna Makaruk
Publisher : World Scientific
Page : 771 pages
File Size : 53,6 Mb
Release : 2023-09-27
Category : Mathematics
ISBN : 9789811271168

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Scientific Legacy Of Professor Zbigniew Oziewicz: Selected Papers From The International Conference "Applied Category Theory Graph-operad-logic" by Hilda Maria Colin Garcia,Jose De Jesus Cruz Guzman,Louis H Kauffman,Hanna Makaruk Pdf

Dedicated to the memory of the late Professor Zbigniew Oziewicz from Universidad Nacional Autónoma de México, the book consists of papers on a wide variety of topics related to the work of Professor Oziewicz, which were presented at the special conference on Graph-Operads-Logic (GOL 2021), selected through peer review to promote his scientific legacy.Professor Oziewicz was a great enthusiast and supporter of category theory and its applications in physics, as well as in various areas of mathematics (topology, noncommutative geometry, etc.). In particular, he made significant contributions to the theory of Frobenius algebras, which now are becoming more important due to their connection with topological quantum field theories that are used in mathematical physics and in quantum topology. Professor Oziewicz was a great and very generous teacher, who immersed his students in the beautiful ideas of category theory as well as mathematical physics and computation. It was his idea to start a series of conferences under the title Graphs-Operads-Logic, most of them held in Mexico, with some of them in the USA, which were a great platform to discuss various ideas connected with category theory and its various applications, and to make friends with other scientists. Despite his passing, the GOL 2021 conference is included in this series to pay tribute to his many contributions to diverse areas of science.The book is laid out in twelve main topics where we can find relevant works from distinguished experts.

The Geometry Of The Universe

Colin Rourke
The Geometry Of The Universe Book PDF

Author : Colin Rourke
Publisher : World Scientific
Page : 274 pages
File Size : 54,6 Mb
Release : 2021-06-03
Category : Science
ISBN : 9789811233883

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The Geometry Of The Universe by Colin Rourke Pdf

Cosmology, the study of the universe, arouses a great deal of public interest, with serious articles both in the scientific press and in major newspapers, with many of the theories and concepts (e.g. the 'big bang' and 'black holes') discussed, often in great depth.Accordingly the book is divided into three parts:Part 1 is readable (and understandable) by anyone with a nodding acquaintance with the basic language of cosmology: events, lights paths, galaxies, black holes and so on. It covers the whole story of the book in a way as untechnical as possible given the scope of the topics covered.Part 2 covers the same ground again but with enough technical details to satisfy a reader with basic knowledge of mathematics and/or physics.Part 3 consists of appendices which are referred to in the other parts and which also contain the highly technical material omitted from Section 2.

Braid and Knot Theory in Dimension Four

Seiichi Kamada
Braid and Knot Theory in Dimension Four Book PDF

Author : Seiichi Kamada
Publisher : American Mathematical Soc.
Page : 329 pages
File Size : 55,9 Mb
Release : 2002
Category : Braid theory
ISBN : 9780821829691

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Braid and Knot Theory in Dimension Four by Seiichi Kamada Pdf

Braid theory and knot theory are related via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus, one can use braid theory to study knot theory and vice versa. In this book, the author generalizes braid theory to dimension four. He develops the theory of surface braids and applies it tostudy surface links. In particular, the generalized Alexander and Markov theorems in dimension four are given. This book is the first to contain a complete proof of the generalized Markov theorem. Surface links are studied via the motion picture method, and some important techniques of this method arestudied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. Included is a table of knotted surfaces with a computation of Alexander polynomials. Braid techniques are extended to represent link homotopy classes. The book is geared toward a wide audience, from graduatestudents to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers.

Braids and Coverings

Vagn Lundsgaard Hansen
Braids and Coverings Book PDF

Author : Vagn Lundsgaard Hansen
Publisher : Cambridge University Press
Page : 208 pages
File Size : 44,9 Mb
Release : 1989-12-07
Category : Mathematics
ISBN : 0521387574

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Braids and Coverings by Vagn Lundsgaard Hansen Pdf

Essays develop the elementary theory of Artin Braid groups geometrically and via homotopy theory, discuss the link between knot theory and the combinatorics of braid groups through Markou's Theorem and investigate polynomial covering maps.

Properties of Closed 3-Braids and Braid Representations of Links

Alexander Stoimenow
Properties of Closed 3 Braids and Braid Representations of Links Book PDF

Author : Alexander Stoimenow
Publisher : Springer
Page : 110 pages
File Size : 43,6 Mb
Release : 2017-12-08
Category : Mathematics
ISBN : 3319681486

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Properties of Closed 3-Braids and Braid Representations of Links by Alexander Stoimenow Pdf

This book studies diverse aspects of braid representations via knots and links. Complete classification results are illustrated for several properties through Xu’s normal 3-braid form and the Hecke algebra representation theory of link polynomials developed by Jones. Topological link types are identified within closures of 3-braids which have a given Alexander or Jones polynomial. Further classifications of knots and links arising by the closure of 3-braids are given, and new results about 4-braids are part of the work. Written with knot theorists, topologists,and graduate students in mind, this book features the identification and analysis of effective techniques for diagrammatic examples with unexpected properties.

The Mathematical Theory of Knots and Braids

S. Moran
The Mathematical Theory of Knots and Braids Book PDF

Author : S. Moran
Publisher : Elsevier
Page : 294 pages
File Size : 47,8 Mb
Release : 2000-04-01
Category : Mathematics
ISBN : 0080871933

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The Mathematical Theory of Knots and Braids by S. Moran Pdf

This book is an introduction to the theory of knots via the theory of braids, which attempts to be complete in a number of ways. Some knowledge of Topology is assumed. Necessary Group Theory and further necessary Topology are given in the book. The exposition is intended to enable an interested reader to learn the basics of the subject. Emphasis is placed on covering the theory in an algebraic way. The work includes quite a number of worked examples. The latter part of the book is devoted to previously unpublished material.

Knots and Links

Peter R. Cromwell
Knots and Links Book PDF

Author : Peter R. Cromwell
Publisher : Cambridge University Press
Page : 356 pages
File Size : 54,5 Mb
Release : 2004-10-14
Category : Mathematics
ISBN : 0521548314

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Knots and Links by Peter R. Cromwell Pdf

A richly illustrated 2004 textbook on knot theory; minimal prerequisites but modern in style and content.

An Interactive Introduction to Knot Theory

Inga Johnson,Allison Henrich
An Interactive Introduction to Knot Theory Book PDF

Author : Inga Johnson,Allison Henrich
Publisher : Courier Dover Publications
Page : 193 pages
File Size : 47,8 Mb
Release : 2017-01-18
Category : Mathematics
ISBN : 9780486804637

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An Interactive Introduction to Knot Theory by Inga Johnson,Allison Henrich Pdf

This well-written and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. The hands-on approach features many exercises to be completed by readers. Prerequisites are only a basic familiarity with linear algebra and a willingness to explore the subject in a hands-on manner. The opening chapter offers activities that explore the world of knots and links — including games with knots — and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.

Knots, Links, Braids, and 3-manifolds

V. V. Prasolov,A. B. Sossinsky
Knots Links Braids and 3 manifolds Book PDF

Author : V. V. Prasolov,A. B. Sossinsky
Publisher : American Mathematical Soc.
Page : 239 pages
File Size : 40,8 Mb
Release : 1997
Category : Science
ISBN : 9780821808986

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Knots, Links, Braids, and 3-manifolds by V. V. Prasolov,A. B. Sossinsky Pdf

This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of 3-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones-Witten invariants are constructed via the rigorous skein algebra approach (mainly due to the Saint Petersburg school). Unlike several recent monographs, where all of these invariants are introduced by using the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites are minimal in this book. Numerous figures and problems make it suitable as a course text and for self-study.