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000845608 1001_ $$aMurakami, Hitoshi,$$eauthor.
000845608 24510 $$aVolume conjecture for knots /$$cHitoshi Murakami, Yoshiyuki Yokota.
000845608 264_1 $$aSingapore :$$bSpringer,$$c2018.
000845608 300__ $$a1 online resource.
000845608 336__ $$atext$$btxt$$2rdacontent
000845608 337__ $$acomputer$$bc$$2rdamedia
000845608 338__ $$aonline resource$$bcr$$2rdacarrier
000845608 4901_ $$aSpringerBriefs in mathematical physics ;$$vvolume 30
000845608 504__ $$aIncludes bibliographical references and index.
000845608 5050_ $$aIntro; Preface; Contents; Acronyms; 1 Preliminaries; 1.1 Knot; 1.2 Satellite; 1.3 Braid; 2 R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant; 2.1 A Link Invariant Derived from a Yang-Baxter Operator; 2.1.1 Yang-Baxter Operator; 2.1.2 Colored Jones Polynomial; 2.1.3 Kashaev's R-Matrix; 2.1.4 Example of Calculation; 2.2 Colored Jones Polynomial via the Kauffman Bracket; 2.2.1 Kauffman Bracket; 2.2.2 Example of Calculation; 3 Volume Conjecture; 3.1 Volume Conjecture; 3.2 Figure-Eight Knot; 3.3 Torus Knot; 4 Idea of ``Proof''; 4.1 Algebraic Part; 4.2 Analytic Part
000845608 5058_ $$a4.2.1 Integral Expression4.2.2 Potential Function; 4.2.3 Saddle Point Method; 4.2.4 Remaining Tasks; 4.3 Geometric Part; 4.3.1 Ideal Triangulation; 4.3.2 Cusp Triangulation; 4.3.3 Hyperbolicity Equations; 4.3.4 Complex Volumes; 5 Representations of a Knot Group, Their Chern-Simons Invariants, and Their Reidemeister Torsions; 5.1 Representations of a Knot Group; 5.1.1 Presentation; 5.1.2 Representation; 5.2 The Chern-Simons Invariant; 5.2.1 Definition; 5.2.2 How to Calculate; 5.3 Twisted SL(2; C) Reidemeister Torsion; 5.3.1 Definition; 5.3.2 How to Calculate
000845608 5058_ $$a6 Generalizations of the Volume Conjecture6.1 Complexification; 6.2 Refinement; 6.2.1 Figure-Eight Knot; 6.2.2 Torus Knot; 6.3 Parametrization; 6.3.1 Torus Knot; 6.3.2 Figure-Eight Knot; 6.4 Miscellaneous Results; 6.5 Final Remarks; References; Index
000845608 506__ $$aAccess limited to authorized users.
000845608 520__ $$aThe volume conjecture states that a certain limit of the colored Jones polynomial of a knot in the three-dimensional sphere would give the volume of the knot complement. Here the colored Jones polynomial is a generalization of the celebrated Jones polynomial and is defined by using a so-called R-matrix that is associated with the N-dimensional representation of the Lie algebra sl(2;C). The volume conjecture was first stated by R. Kashaev in terms of his own invariant defined by using the quantum dilogarithm. Later H. Murakami and J. Murakami proved that Kashaev’s invariant is nothing but the N-dimensional colored Jones polynomial evaluated at the Nth root of unity. Then the volume conjecture turns out to be a conjecture that relates an algebraic object, the colored Jones polynomial, with a geometric object, the volume. In this book we start with the definition of the colored Jones polynomial by using braid presentations of knots. Then we state the volume conjecture and give a very elementary proof of the conjecture for the figure-eight knot following T. Ekholm. We then give a rough idea of the “proof”, that is, we show why we think the conjecture is true at least in the case of hyperbolic knots by showing how the summation formula for the colored Jones polynomial “looks like” the hyperbolicity equations of the knot complement. We also describe a generalization of the volume conjecture that corresponds to a deformation of the complete hyperbolic structure of a knot complement. This generalization would relate the colored Jones polynomial of a knot to the volume and the Chern–Simons invariant of a certain representation of the fundamental group of the knot complement to the Lie group SL(2;C). We finish by mentioning further generalizations of the volume conjecture.--$$cProvided by publisher.
000845608 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed August 20, 2018).
000845608 650_0 $$aKnot theory.
000845608 7001_ $$aYokota, Yoshiyuki,$$eauthor.
000845608 77608 $$iPrint version: $$z9811311498$$z9789811311499$$w(OCoLC)1037281776
000845608 830_0 $$aSpringerBriefs in mathematical physics ;$$vv. 30.
000845608 852__ $$bebk
000845608 85640 $$3SpringerLink$$uhttps://univsouthin.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-981-13-1150-5$$zOnline Access$$91397441.1
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