Lecture Notes in Mathematics

Guts of Surfaces and the Colored Jones Polynomial

Authors: Futer, David, Kalfagianni, Efstratia, Purcell, Jessica

  • 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
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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)


Table of contents (10 chapters)

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-642-33302-6
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA
  • ISBN 978-3-642-33301-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Guts of Surfaces and the Colored Jones Polynomial
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2069
Copyright
2013
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-33302-6
DOI
10.1007/978-3-642-33302-6
Softcover ISBN
978-3-642-33301-9
Series ISSN
0075-8434
Edition Number
1
Number of Pages
X, 170
Number of Illustrations and Tables
17 b/w illustrations, 45 illustrations in colour
Topics