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E-book
Author Futer, David

Title Guts of surfaces and the colored Jones polynomial / David Futer, Efstratia Kalfagianni, Jessica Purcell
Published Berlin : Springer, ©2013

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Description 1 online resource (x, 170 pages) : illustrations (some color)
Series Lecture notes in mathematics, 1617-9692 ; 2069
Lecture notes in mathematics (Springer-Verlag) ; 2069
Contents Introduction -- Decomposition into 3-Balls -- Ideal Polyhedra -- I-Bundles and Essential Product Disks -- Guts and Fibers -- Recognizing Essential Product Disks -- Diagrams Without Non-prime Arcs -- Montesinos Links -- Applications -- Discussion and Questions
Summary This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the complement to the combinatorics of certain surface spines (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our method bridges the gap between quantum and geometric knot invariants
Bibliography Includes bibliographical references and index
Notes English
Online resource; title from PDF title page (SpringerLink, viewed December 24, 2012)
Subject Complex manifolds.
Geometry, Hyperbolic.
Variedades complejas
Geometría hiperbólica
Complex manifolds
Geometry, Hyperbolic
Knoten Mathematik
Niederdimensionale Topologie
Form Electronic book
Author Kalfagianni, Efstratia
Purcell, Jessica
ISBN 3642333028
9783642333026