CaltechAUTHORS: Article
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 02 Aug 2024 18:55:26 -0700The virtual Haken conjecture: Experiments and examples
https://resolver.caltech.edu/CaltechAUTHORS:DUNgt03
Year: 2003
DOI: 10.2140/gt.2003.7.399
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture.
First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem.
Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.https://resolver.caltech.edu/CaltechAUTHORS:DUNgt03Non-triviality of the A-polynomial for knots in S³
https://resolver.caltech.edu/CaltechAUTHORS:DUNagt04
Year: 2004
DOI: 10.2140/agt.2004.4.1145
The A-polynomial of a knot in S³ defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL₂C. Here, we show that a non-trivial knot in S³ has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU₂-representations of Dehn surgeries on knots in S³. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot.https://resolver.caltech.edu/CaltechAUTHORS:DUNagt04An ascending HNN extension of a free group inside SL_2 ℂ
https://resolver.caltech.edu/CaltechAUTHORS:20100318-151445653
Year: 2006
DOI: 10.1090/S0002-9939-06-08398-5
We give an example of a subgroup of SL_2 ℂ which is a strictly
ascending HNN extension of a non-abelian finitely generated free group F. In
particular, we exhibit a free group F in SL_2 ℂ of rank 6 which is conjugate
to a proper subgroup of itself. This answers positively a question of Drutu
and Sapir (2005). The main ingredient in our construction is a specific finite
volume (non-compact) hyperbolic 3-manifold M which is a surface bundle over
the circle. In particular, most of F comes from the fundamental group of a
surface fiber. A key feature of M is that there is an element of π1(M) in SL_2 ℂ
with an eigenvalue which is the square root of a rational integer. We also use
the Bass-Serre tree of a field with a discrete valuation to show that the group
F we construct is actually free.https://resolver.caltech.edu/CaltechAUTHORS:20100318-151445653The Superpolynomial for Knot Homologies
https://resolver.caltech.edu/CaltechAUTHORS:20160511-091653426
Year: 2006
DOI: 10.1080/10586458.2006.10128956
We propose a framework for unifying the sl(N) Khovanov– Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory that categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large-N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee.
While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture that gives new predictions even for the original sl(2) Khovanov homology.https://resolver.caltech.edu/CaltechAUTHORS:20160511-091653426Automorphic forms and rational homology 3–spheres
https://resolver.caltech.edu/CaltechAUTHORS:CALgt06b
Year: 2006
DOI: 10.2140/gt.2006.10.295
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3–spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3–manifold. The conjectures
we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL2 of an imaginary quadratic field.
The proof of this theorem involves ruling out the existence of an irreducible two dimensional Galois representation rho of Gal(Qbar/Qsqrt-2) satisfying certain prescribed ramification conditions. In contrast to similar questions of this form, rho is allowed to have arbitrary ramification at some prime pi of Z[sqrt -2].
In the next paper in this volume, Boston and Ellenberg apply pro–p techniques to our examples and show that our result is true unconditionally. Here, we give additional examples where their techniques apply, including some non-arithmetic examples.
Finally, we investigate the congruence covers of twist-knot orbifolds. Our experimental evidence suggests that these topologically similar orbifolds have rather different behavior depending on whether or not they are arithmetic. In particular, the congruence covers of the non-arithmetic orbifolds have a paucity of homology.https://resolver.caltech.edu/CaltechAUTHORS:CALgt06bA random tunnel number one 3–manifold does not fiber over the circle
https://resolver.caltech.edu/CaltechAUTHORS:DUNgt06
Year: 2006
DOI: 10.2140/gt.2006.10.2431
We address the question: how common is it for a 3–manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3–manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3–manifolds.
The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3–manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem.
The 3–manifold situation contrasts markedly with random 2–generator 1–relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.https://resolver.caltech.edu/CaltechAUTHORS:DUNgt06Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds
https://resolver.caltech.edu/CaltechAUTHORS:20100218-101507964
Year: 2010
DOI: 10.1353/ajm.0.0098
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration M → S^1 is the modular elliptic curve E = X_0(49), which admits multiplication by the ring of integers of Q[√(−7)]. We first base change the holomorphic
differential on E to a cusp form on GL(2) over K = Q[√(−3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of O^∗_D of level 7. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.https://resolver.caltech.edu/CaltechAUTHORS:20100218-101507964