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Knots and Links in Three-Dimensional Flows
The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed.
224 pages, Paperback
First published April 18, 1997
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September 25, 2017
On the whole, this is a nicely-conceived book on knots and dynamics (in particular, knots and links that occur in flows satisfying certain hyperbolicity conditions).
I have some complaints about it, primarily the handwaviness at some parts (e.g. (i) I don't like the way branched manifolds were introduced - only branched 1-manifolds were defined and it was done in a very handwavey manner and it wasn't clear from this monograph how one should think about branched 2-manifolds, which were relevant to template theory, and how e.g. the collapsing map respects periodic orbits contained within a suitable compact neighbourhood by isotoping them onto a branched manifold; (ii) Pitchfork bifurcation wasn't properly defined - Ghrist et. al. simply remarked that it arose naturally in systems that were invariant under the transformation x |--> -x. Well, this made things unclear later on since other bifurcations may also arise in systems that invariant under the above symmetry transformation (e.g. period-doubling bifurcation), which was problematic in the case of Lemma 2.2.11, when it was argued that a particular Poincare map induced by a 3-d hyperbolic flow might undergo a pitchfork bifurcation or a period-doubling bifurcation depending on the orientation of the map - the fact that we weren't given the sufficient conditions for a pitchfork bifurcation in this particular case made the explanation rather unconvincing.) and some rather puzzling logical moves (e.g. Proposition 2.2.12's explanation of why Λ is an expanding attractor - why is it that if W^u(γ) is dense in the 3-dimensional basic set B, the dimension of Λ is less than or equal to 2? We know that Λ is the complement of W^u(γ) but so what? As far as I can see, just because W^u(γ) is dense in B, it does not mean that it has dimension of at least 1. For instance, Q^3 is dense in R^3 yet has topological dimension of 0.)
These complaints aren't trivial ones, and they did annoy me a great deal, however I still think it's important to appreciate other aspects of the book. For one, the book is quite well-organised - I like how it started with a crash course into knot and link theory & dynamical theory (though I think the whole part about Bowen's theorem on subshifts of finite types and rectangles wasn't done too well ... see Franks' Homology and Dynamical Systems for a better exposition), and how it motivated Birman-Williams' Template Theory and actually proved it (this book and the original 1983 paper by Birman-Williams are the only two sources I can find that actually prove this theorem. It was nice to have another perspective on how this proof works because Birman-Williams' paper skipped a lot of details.) and how this opened up the discussion to other aspects of Template theory. This book is very good in its discussion of general template theory, particularly in discussing Ghrist (and his coworker's work) on proving that there exists a universal template containing all knots and links (with finite crossings).
I found this book a little tough-going in the middle because I was new to this subject, but there's much to enjoy here, and it opens up a range of other very interesting questions worth thinking about - in particular, how else may we understand the behaviour of dynamical systems from a topological point of view? All in all, though there are parts of the book I have issues with, I'm very glad it exists. 3.5 stars but rounding up!
I have some complaints about it, primarily the handwaviness at some parts (e.g. (i) I don't like the way branched manifolds were introduced - only branched 1-manifolds were defined and it was done in a very handwavey manner and it wasn't clear from this monograph how one should think about branched 2-manifolds, which were relevant to template theory, and how e.g. the collapsing map respects periodic orbits contained within a suitable compact neighbourhood by isotoping them onto a branched manifold; (ii) Pitchfork bifurcation wasn't properly defined - Ghrist et. al. simply remarked that it arose naturally in systems that were invariant under the transformation x |--> -x. Well, this made things unclear later on since other bifurcations may also arise in systems that invariant under the above symmetry transformation (e.g. period-doubling bifurcation), which was problematic in the case of Lemma 2.2.11, when it was argued that a particular Poincare map induced by a 3-d hyperbolic flow might undergo a pitchfork bifurcation or a period-doubling bifurcation depending on the orientation of the map - the fact that we weren't given the sufficient conditions for a pitchfork bifurcation in this particular case made the explanation rather unconvincing.) and some rather puzzling logical moves (e.g. Proposition 2.2.12's explanation of why Λ is an expanding attractor - why is it that if W^u(γ) is dense in the 3-dimensional basic set B, the dimension of Λ is less than or equal to 2? We know that Λ is the complement of W^u(γ) but so what? As far as I can see, just because W^u(γ) is dense in B, it does not mean that it has dimension of at least 1. For instance, Q^3 is dense in R^3 yet has topological dimension of 0.)
These complaints aren't trivial ones, and they did annoy me a great deal, however I still think it's important to appreciate other aspects of the book. For one, the book is quite well-organised - I like how it started with a crash course into knot and link theory & dynamical theory (though I think the whole part about Bowen's theorem on subshifts of finite types and rectangles wasn't done too well ... see Franks' Homology and Dynamical Systems for a better exposition), and how it motivated Birman-Williams' Template Theory and actually proved it (this book and the original 1983 paper by Birman-Williams are the only two sources I can find that actually prove this theorem. It was nice to have another perspective on how this proof works because Birman-Williams' paper skipped a lot of details.) and how this opened up the discussion to other aspects of Template theory. This book is very good in its discussion of general template theory, particularly in discussing Ghrist (and his coworker's work) on proving that there exists a universal template containing all knots and links (with finite crossings).
I found this book a little tough-going in the middle because I was new to this subject, but there's much to enjoy here, and it opens up a range of other very interesting questions worth thinking about - in particular, how else may we understand the behaviour of dynamical systems from a topological point of view? All in all, though there are parts of the book I have issues with, I'm very glad it exists. 3.5 stars but rounding up!
Displaying 1 of 1 review