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Graphs on Surfaces: Dualities, Polynomials, and Knots (Springerbriefs in Mathematics)

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“Here, the venerable knot-theoretic and graph-theoretic themes find a host of unifying common generalizations. Undergraduates will appreciate the patient and visual development of the foundations, particularly the dualities (paired representations of a single structure). Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (D. V. Feldman, Choice, Vol. 51 (7), March, 2014)

“This monograph is aimed at researchers both in graph theory and in knot theory. It should be accessible to a graduate student with a grounding in both subjects. There are (colour) diagrams throughout. … The monograph gives a unified treatment of various ideas that have been studied and used previously, generalising many of them in the process.” (Jessica Banks, zbMATH, Vol. 1283, 2014)

“The authors have composed a very interesting and valuable work. … For properly prepared readers … the book under review is the occasion for all sorts of fun including the inner life of ribbon groups, Tait graphs, Penrose polynomials, Tutte polynomials, and of course Jones polynomials and HOMFLY polynomials. This is fascinating mathematics, presented in a clear and accessible way.” (Michael Berg, MAA Reviews, October, 2013)

--This text refers to an alternate Paperback edition.
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Product Details

  • Series: Springerbriefs in Mathematics
  • Paperback: 149 pages
  • Publisher: Not Avail
  • ISBN-10: 1461469716
  • ISBN-13: 978-1461469711
  • Shipping Weight: 1.7 pounds
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)

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Format: Paperback
I'm not a mathematician and no longer a student, just someone who reads math occasionally for refreshment. I picked up this book as part of my summer break reading, and found it perfectly suited to an enjoyable couple of days. Although it claims to be directed to an audience of grad students and professionals, I'm far away from either qualification: having read a book or two about knots and pieces of books about graphs in the distant past was enough to enable me to enjoy most of it.

Most of the book relates to material published jointly or severally by the authors in the past couple of years, especially concerning the group action of duals and twists on ribbon graphs. Many of the pertinent references are on the arXiv and more than a few date from between 2010-2014, so the material is still very current as I write. However, the book is a synthesis and refinement, not just a repetition, of that material. It's richly illustrated, so you can learn a lot from trying to reproduce the diagrams. The pictures alone are already very suggestive of how embedded graphs are connected to knots and links. (I also really liked the authors' phrase "twisted duality" -- geometrically apt, but maybe also a useful metaphor for marriage counselors and other therapists?)

I do recommend though that you know something about graph and knot polynomials, which are of great interest to working mathematicians. My knowledge was more than zero but epsilon-sized in this area, and I wound up skimming large chunks of the algebraic proofs showing how various polynomials (e.g. Tutte, Penrose, Bollabás-Riordan, Jones, and HOMFLY, plus Kaufmann bracket) were related to each other and/or to the authors' "transition polynomial".
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