**Author**: J. P. Levine

**Publisher:** Springer

**ISBN:** 3540097392

**Category:** Mathematics

**Page:** 110

**View:** 174

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Classical circuit theory is a mathematical theory of linear, passive circuits, namely, circuits composed of resistors, capacitors and inductors. Like many a thing classical, it is old and enduring, structured and precise, simple and elegant. It is simple in that everything in it can be deduced from ?rst principles based on a few physical laws. It is enduring in that the things we can say about linear, passive circuits are universally true, unchanging. No matter how complex a circuit may be, as long as it consists of these three kinds of elements, its behavior must be as prescribed by the theory. The theory tells us what circuits can and cannot do. As expected of any good theory, classical circuit theory is also useful. Its ulti mate application is circuit design. The theory leads us to a design methodology that is systematic and precise. It is based on just two fundamental theorems: that the impedance function of a linear, passive circuit is a positive real function, and that the transfer function is a bounded real function, of a complex variable.
Knot theory is a rapidly developing field of research with many applications, not only for mathematics. The present volume, written by a well-known specialist, gives a complete survey of this theory from its very beginnings to today's most recent research results. An indispensable book for everyone concerned with knot theory.
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 to study 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 are studied.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 graduate students to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers.
Bringing together many results previously scattered throughout the research literature into a single framework, this work concentrates on the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory.
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 OCo 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.
Skein modules, as invariants of 3-manifolds, were introduced by J. Przytycki and V. Turaev in 1987 and have since become a popular area of research investigations. In particular, skein modules provide sound algebraic structures for studying 3-dimensional manifolds and knot theory in 3-manifolds. In most cases, skein modules carry an additional algebraic structure such as Hopf algebra, Lie algebra, coordinate ring of SL (2, [Special characters omitted.] ) - character variety of representations of the fundamental group of a 3-manifold M3, etc. The Kauffman Bracket Skein Module (KBSM) is the most extensively studied skein module. Computing and understanding the structure of the KBSM is usually a challenging task even for 3-manifolds with simple geometric structure. In a recent paper M. Dabkowski and M. Mroczkowski computed the KBSM for F0,3 x S1 ; ( F0,3 denotes the surface of genus 0 with 3 boundary components) and showed that the KBSM for F0,3 x S1 is free. In this dissertation we outline the method of calculation for the KBSM of the manifold M3 that is obtained from the disk sum of A x S1 and A x I, (where A is an annulus) and demonstrate how to extend the method used by Dabkowski and Mroczkowski to this manifold.
This book presents the relationship between classical theta functions and knots. It is based on a novel idea of Razvan Gelca and Alejandro Uribe, which converts Weil''s representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology. Theta Functions and Knots can be read in two perspectives. People with an interest in theta functions or knot theory can learn how the two are related. Those interested in ChernOCoSimons theory find here an introduction using the simplest case, that of abelian ChernOCoSimons theory. Moreover, the construction of abelian ChernOCoSimons theory is based entirely on quantum mechanics, and not on quantum field theory as it is usually done. Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is a self-contained, unified presentation. It is suitable for an advanced graduate course, as well as for self-study. Contents: Some Historical Facts; A Quantum Mechanical Prototype; Surfaces and Curves; The Theta Functions Associated to a Riemann Surface; From Theta Functions to Knots; Some Results About 3- and 4-Dimensional Manifolds; The Discrete Fourier Transform and Topological Quantum Field Theory; Theta Functions and Quantum Groups; An Epilogue OCo Abelian ChernOCoSimons Theory. Readership: Graduate students and young researchers with an interest in complex analysis, mathematical physics, algebra geometry and low dimensional topology.
From prehistory to the present, knots have been used for purposes both artistic and practical. The modern science of Knot Theory has ramifications for biochemistry and mathematical physics and is a rich source of research projects for undergraduate and graduate students and professionals alike. Quandles are essentially knots translated into algebra. This book provides an accessible introduction to quandle theory for readers with a background in linear algebra. Important concepts from topology and abstract algebra motivated by quandle theory are introduced along the way. With elementary self-contained treatments of topics such as group theory, cohomology, knotted surfaces and more, this book is perfect for a transition course, an upper-division mathematics elective, preparation for research in knot theory, and any reader interested in knots.

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**Author**: J. P. Levine

**Publisher:** Springer

**ISBN:** 3540097392

**Category:** Mathematics

**Page:** 110

**View:** 174

*[G] M. Gutierrez : On Knot Modules, Invent. Math. 17 (1972) , 329 – 35. [Hil F. Hirzebruch, W. Neumann, S. Koh : Differentiable Manifolds and Quadratic Forms. New York: Dekker, 1971. [JJ R. Jacobowitz: Hermitian Forms over Global Fields ...*

**Author**: J. P. Levine

**Publisher:** Springer

**ISBN:** 9783540385554

**Category:** Mathematics

**Page:** 110

**View:** 957

*772: J. P. Levine, Algebraic Structure of Knot Modules x1, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Ete de ...*

**Author**: J. Krzyz

**Publisher:** Springer

**ISBN:** 3540119892

**Category:** Mathematics

**Page:** 184

**View:** 899

*Classical circuit theory is a mathematical theory of linear, passive circuits, namely, circuits composed of resistors, capacitors and inductors.*

**Author**: Omar Wing

**Publisher:** Springer

**ISBN:** 0387097392

**Category:** Technology & Engineering

**Page:** 296

**View:** 832

*(1980) Algebraic structure of knot modules, Lect. Notes in Math., 772, Springer Verlag. (1982) The module of a 2-component link, Comment. Math. Helv., 57: 377–399. (1983) Doubly slice knots and doubled disk knots, Michigan J. Math., ...*

**Author**: Akio Kawauchi

**Publisher:** Birkhäuser

**ISBN:** 9783034892278

**Category:** Mathematics

**Page:** 423

**View:** 825

*229(1977) 1–50 [475] J. Levine, Some results on higher dimensional knot groups, in “Knot Theory” (Switzerland, 1977), Lect. Notes in Math., 685, Springer Verlag (1978) 243–269 [476] J. Levine, Algebraic structure of knot modules, Lect.*

**Author**: Seiichi Kamada

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821829691

**Category:** Mathematics

**Page:** 313

**View:** 586

*8, 98–110 (1969) [159] --, An algebraic classification of some knots of codimension two, Comm. Math. Helv. 45, 185–198 (1970) [160] --, Algebraic structure of knot modules, Lecture Notes in Mathematics 772, Springer (1980) [161] -- and ...*

**Author**: Andrew Ranicki

**Publisher:** Springer Science & Business Media

**ISBN:** 9783662120118

**Category:** Mathematics

**Page:** 646

**View:** 887

*[Lev] Levine, J. Algebraic Structure of Knot Modules, Lecture Notes in Mathematics 772, Springer-Verlag, Berlin - Heidelberg - New York (1980). [Lom] Lomonaco, S.J., Jr, (editor) Low Dimensional Topology, Contemporary Mathematics 20, ...*

**Author**: Jonathan Arthur Hillman

**Publisher:** World Scientific

**ISBN:** 9789814407397

**Category:** Mathematics

**Page:** 353

**View:** 756

*Skein modules, as invariants of 3-manifolds, were introduced by J. Przytycki and V. Turaev in 1987 and have since become a popular area of research investigations.*

**Author**: Billye E. Cheek

**Publisher:**

**ISBN:** OCLC:761204404

**Category:** Knot theory

**Page:** 192

**View:** 955

*R. L. Epstein, Degrees of Unsolvability: Structure and Theory, XIV, 216 pages. ... R. S. Doran and J. Wichmann, Approximate laentities and Factorization in Banach Modules. ... J. P. Levine, Algebraic Structure of Knot Modules.*

**Author**: A. Libgober

**Publisher:** Springer

**ISBN:** 9783540387206

**Category:** Mathematics

**Page:** 288

**View:** 511

*772: J. P. Levine, Algebraic Structure of Knot Modules. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...*

**Author**: R. Keith Dennis

**Publisher:** Springer

**ISBN:** 9783540395560

**Category:** Mathematics

**Page:** 412

**View:** 579

*772: J. P. Levine, Algebraic Structure of Knot Modules. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979, Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...*

**Author**: Eric Friedlander

**Publisher:** Springer

**ISBN:** 9783540386469

**Category:** Mathematics

**Page:** 524

**View:** 822

*772: J. P. Levine, Algebraic Structure of Knot Modules. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...*

**Author**: F. van Oystaeyen

**Publisher:** Springer

**ISBN:** 9783540390572

**Category:** Mathematics

**Page:** 300

**View:** 517

*772: J. P. Levine, Algebraic Structure of Knot Modules. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...*

**Author**: R. Keith Dennis

**Publisher:** Springer

**ISBN:** 9783540395539

**Category:** Mathematics

**Page:** 414

**View:** 127

*Skein modules are constructed by the general method that governs algebraic topology: start with a large algebraic structure produced using topological objects, factor it by algebraic relations motivated by topological properties and ...*

**Author**: R?zvan Gelca

**Publisher:** World Scientific

**ISBN:** 9789814520584

**Category:** Mathematics

**Page:** 468

**View:** 992

*772: J. P. Levine, Algebraic Structure of Knot Modules. XI, 104 pages, 1980, Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages, 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...*

**Author**: V. Dlab

**Publisher:** Springer

**ISBN:** 9783540383871

**Category:** Mathematics

**Page:** 676

**View:** 311

**Algebraic Structure of Knot Modules**. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...

**Author**: E. J. Billington

**Publisher:** Springer

**ISBN:** 9783540393757

**Category:** Mathematics

**Page:** 446

**View:** 251

*Quandles are essentially knots translated into algebra. This book provides an accessible introduction to quandle theory for readers with a background in linear algebra.*

**Author**: Mohamed Elhamdadi

**Publisher:** American Mathematical Soc.

**ISBN:** 9781470422134

**Category:** Knot theory

**Page:** 245

**View:** 638

**Algebraic Structure of Knot Modules**. XI, 104 pages. 1980. Vol. 773: Numerical Analysis. Proceedings, 1979. Edited by G. A. Watson. X, 184 pages. 1980. Vol. 774: R. Azencott, Y. Guivarc'h, R. F. Gundy, Ecole d'Eté de ...

**Author**: D. Jungnickel

**Publisher:** Springer

**ISBN:** 9783540393801

**Category:** Mathematics

**Page:** 332

**View:** 691

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