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Guts of Surfaces and the Colored Jones Polynomial

  • David Futer
  • Efstratia Kalfagianni
  • Jessica Purcell

Part of the Lecture Notes in Mathematics book series (LNM, volume 2069)

Table of contents

  1. Front Matter
    Pages i-x
  2. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 1-15
  3. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 17-33
  4. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 35-51
  5. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 53-72
  6. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 73-90
  7. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 91-108
  8. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 109-118
  9. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 119-138
  10. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 139-154
  11. David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 155-161
  12. Back Matter
    Pages 163-170

About this book

Introduction

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.

Keywords

57N10, 57M25, 57M27, 57M50, 57M15, 57R56 colored Jones polynomial fiber guts of surface hyperbolic volume

Authors and affiliations

  • David Futer
    • 1
  • Efstratia Kalfagianni
    • 2
  • Jessica Purcell
    • 3
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsBrigham Young UniversityProvoUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-33302-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2013
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-33301-9
  • Online ISBN 978-3-642-33302-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site
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