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Guts of Surfaces and the Colored Jones Polynomial

  • Book
  • © 2013

Overview

Authors:
  1. David Futer

    1. Department of Mathematics, Temple University, Philadelphia, USA

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  2. Efstratia Kalfagianni

    1. Department of Mathematics, Michigan State University, East Lansing, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Jessica Purcell

    1. Department of Mathematics, Brigham Young University, Provo, USA

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  • 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
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  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
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Keywords

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.

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

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