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  • Book
  • © 2013

Guts of Surfaces and the Colored Jones Polynomial

  • Relates all central areas of modern 3-dimensional topology

  • The first monograph which initiates a systematic study of relations between quantum and geometric topology

  • Appeals to a broad audience of 3-dimensional topologists: combines tools from mainstream areas of 3-dimensional topology

  • Includes supplementary material: sn.pub/extras

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

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
  2. Introduction

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 1-15
  3. Decomposition into 3-Balls

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 17-33
  4. Ideal Polyhedra

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 35-51
  5. I-Bundles and Essential Product Disks

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 53-72
  6. Guts and Fibers

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 73-90
  7. Recognizing Essential Product Disks

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 91-108
  8. Diagrams Without Non-prime Arcs

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 109-118
  9. Montesinos Links

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 119-138
  10. Applications

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 139-154
  11. Discussion and Questions

    • David Futer, Efstratia Kalfagianni, Jessica Purcell
    Pages 155-161
  12. Back Matter

    Pages 163-170

About this book

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

Reviews

From the reviews:

 “A relationship between the geometry of knot complements and the colored Jones polynomial is given in this monograph. The writing is well organized and comprehensive, and the book is accessible to both researchers and graduate students with some background in geometric topology and Jones-type invariants.” (Heather A. Dye, Mathematical Reviews, January, 2014)

Authors and Affiliations

  • Department of Mathematics, Temple University, Philadelphia, USA

    David Futer

  • Department of Mathematics, Michigan State University, East Lansing, USA

    Efstratia Kalfagianni

  • Department of Mathematics, Brigham Young University, Provo, USA

    Jessica Purcell

Bibliographic Information

  • Book Title: Guts of Surfaces and the Colored Jones Polynomial

  • Authors: David Futer, Efstratia Kalfagianni, Jessica Purcell

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-33302-6

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2013

  • Softcover ISBN: 978-3-642-33301-9Published: 18 December 2012

  • eBook ISBN: 978-3-642-33302-6Published: 18 December 2012

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: X, 170

  • Number of Illustrations: 17 b/w illustrations, 45 illustrations in colour

  • Topics: Manifolds and Cell Complexes, Hyperbolic Geometry

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions