Guts of surfaces and the colored Jones polynomial David Futer, Efstratia Kalfagianni, Jessica Purcell

Format:

Book

Author/Creator:

Futer, David

Contributor:

Kalfagianni, Efstratia

Purcell, Jessica

Series:

Lecture notes in mathematics (Springer-Verlag) 2069

Lecture notes in mathematics 1617-9692 2069

Language:

English

Subjects (All):

Complex manifolds.

Geometry, Hyperbolic.

Physical Description:

1 online resource

Place of Publication:

Berlin Springer ©2013

Language Note:

English

System Details:

text file

PDF

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

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

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed December 24, 2012)

Other Format:

Printed edition:

ISBN:

3642333028

9783642333026

OCLC:

822868959

Access Restriction:

Restricted for use by site license