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2013 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Guts of Surfaces and the Colored Jones Polynomial

verfasst von: David Futer, Efstratia Kalfagianni, Jessica Purcell

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In the last 3 decades, there has been significant progress in 3-dimensional topology, due in large part to the application of new techniques from other areas of mathematics and from physics. On the one hand, ideas from geometry have led to geometric decompositions of 3-manifolds and to invariants such as the A-polynomial and hyperbolic volume. On the other hand, ideas from quantum physics have led to the development of invariants such as the Jones polynomial and colored Jones polynomials.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 2. Decomposition into 3-Balls
Abstract
In this chapter, we start with a connected link diagram and explain how to construct state graphs and state surfaces. We cut the link complement in S 3 along the state surface, and then describe how to decompose the result into a collection of topological balls whose boundaries have a checkerboard coloring.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 3. Ideal Polyhedra
Abstract
Recall that \({M}_{A} = {S}^{3}\setminus \setminus {S}_{A}\) is S 3 cut along the surface S A . In the last chapter, starting with a link diagram D(K), we obtained a prime decomposition of M A into 3-balls. One of our goals in this chapter is to show that, if D(K) is A-adequate (see Definition 1.1 on p. 4), each of these balls is a checkerboard colored ideal polyhedron with 4-valent vertices.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 4. I-Bundles and Essential Product Disks
Abstract
Recall that we are trying to relate geometric and topological aspects of the knot complement \({S}^{3} \setminus K\) to quantum invariants and diagrammatic properties. So far, we have identified an essential state surface S A , and we have found a polyhedral decomposition of \({M}_{A}\,=\,{S}^{3}\setminus \setminus {S}_{A}\).
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 5. Guts and Fibers
Abstract
This chapter contains one of the main results of the manuscript, namely a calculation of the Euler characteristic of the guts of M A in Theorem 5.14. The calculation will be in terms of the number of essential product disks (EPDs) for M A which are complex, as in Definition 5.2, below.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 6. Recognizing Essential Product Disks
Abstract
Theorem 5.14 reduces the problem of computing the Euler characteristic of the guts of M A to counting how many complex EPDs are required to span the I-bundle of the upper polyhedron. Our purpose in this chapter is to recognize such EPDs from the structure of the all-A state graph \({\mathbb{G}}_{A}\). The main result is Theorem 6.4, which describes the basic building blocks for such EPDs.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 7. Diagrams Without Non-prime Arcs
Abstract
In this chapter, which is independent from the remaining chapters, we will restrict ourselves to A-adequate diagrams D(K) for which the polyhedral decomposition includes no non-prime arcs or switches. In this case, one can simplify the statement of Theorem 5.14 and give an easier combinatorial estimate for the guts of M A . This is done in Theorem 7.2, whose proof takes up the bulk of the chapter.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 8. Montesinos Links
Abstract
In this chapter, we study state surfaces of Montesinos links, and calculate their guts. Our main result is Theorem 8.6. In that theorem, we show that for every sufficiently complicated Montesinos link K, either K or its mirror image admits an A-adequate diagram D such that the quantity \(\vert \vert {E}_{c}\vert \vert \) of Definition 5.9 vanishes.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 9. Applications
Abstract
In this chapter, we will use the calculations of \(\mathrm{guts}({S}^{3}\setminus \setminus {S}_{A})\) obtained in earlier chapters to relate the geometry of A-adequate links to diagrammatic quantities and to Jones polynomials. In Sect. 9.1, we combine Theorem 5.14 with results of Agol et al. [6] to obtain bounds on the volumes of hyperbolic A-adequate links. A sample result is Theorem 9.7, which gives tight diagrammatic estimates on the volumes of positive braids with at least 3 crossings per twist region. The gap between the upper and lower bounds on volume is a factor of about 4.15.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Chapter 10. Discussion and Questions
Abstract
In this final chapter, we state some questions that arose from this work and speculate about future directions related to this project. In Sect. 10.1, we discuss modifications of the diagram D that preserve A-adequacy. In Sect. 10.2, we speculate about using normal surface theory in our polyhedral decomposition of M A to attack various open problems, for example the Cabling Conjecture and the determination of hyperbolic A-adequate knots.
David Futer, Efstratia Kalfagianni, Jessica Purcell
Backmatter
Metadaten
Titel
Guts of Surfaces and the Colored Jones Polynomial
verfasst von
David Futer
Efstratia Kalfagianni
Jessica Purcell
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-33302-6
Print ISBN
978-3-642-33301-9
DOI
https://doi.org/10.1007/978-3-642-33302-6

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