Easy cases of the volume conjecture?
November 18, 2013 mathematics
The volume conjecture relates the hyperbolic volume of a knot complement to quantum invariants of the knot. Specifically, the conjecture is that
When and one takes the limit, this sum transforms quite nicely into the Riemann sum which computes , which is the hyperbolic volume of the figure eight knot complement.
Are there other cases in which one can verify the volume conjecture directly by finding a nice form for the colored Jones polynomial? Yamazaki-Yokota in “On the limit of the colored Jones polynomial of a non-simple link” verified, in the same direct way, the volume conjecture for a certain link with volume . I hope one could find other examples by searching for links and knots with “nice” volumes in terms of the Lobachevsky function .From my search with SnapPy,
- the link in Rolfsen’s table has volume ,
- the link in Rolfsen’s table has volume , and
- the knot volume .
Of course, there are lots of nonhyperbolic knots and it is hard to prove the volume conjecture even for those, in spite of the fact that their hyperbolic volume is “really nice” (namely, zero!).
Many more Lights Out
July 17, 2010 mathematics
A very long while ago I posted some solutions to Lights Out; back then, I solved the -by- board by row-reducing an -by- matrix.
Since then, both Boris Okun and Brent Werness pointed out to me that I should’ve solved Lights Out by using a scanning algorithm: propagating the button presses down one row at a time, and exponentiating the propagation matrix to make sure that I don’t get stuck at the last row.
This is much faster.
With this method, here is a (scaled down, auto-leveled) 2000-by-2000 solution:
And here is a (very much scaled-down, auto-leveled) 5000-by-5000 solution:
Projector on Blackboard
January 19, 2010 mathematics
I recently gave a beamer talk, which gave me the chance to point the beamer at my blackboard.
Classifying manifolds is impossible.
February 12, 2007 mathematics
At a recent Pizza Seminar, Matt Day gave a lovely talk explaining why it isn’t possible to classify 4-manifolds.
An algorithm for deciding whether two closed 4-manifolds are homeomorphic gives an algorithm for deciding whether a closed 4-manifold is simply connected, and therefore (since every finitely presented group is the fundamental group of a 4-manifold), and algorithm for deciding when a group is trivial. Here’s the reduction: we are given a 4-manifold , and we compute its signature . By Freedman, there are no more than two closed simply connected 4-manifolds, and , having the same signature as ; we construct and , and we use the homeomorphism decision procedure to test if or .
Since there is no algorithm for deciding when a group is trivial, there can not be an algorithm for deciding when two closed 4-manifolds are homeomorphic.
There is a paper  discussing some of these issues. In particular, that paper discusses Novikov’s proof that cannot be recognized for .
 A.V. Chernavsky, V.P. Leksine, Unrecognizability of manifolds, Ann. Pure Appl. Logic. 141 (2006) 325–335. https://doi.org/10.1016/j.apal.2005.12.011.
Subgroups of products versus products of subgroups.
February 4, 2007 mathematics
This is a question I wandered into accidentally years ago now, which I think other people might be amused to think about (or more likely, put on an abstract algebra exam).
Let be a group, and a subgroup of . Is always isomorphic to , for some subgroups ? But beware!–I am not requiring (or expecting) any canonicity or naturality for the isomorphism: for instance sits diagonally in , and it just so happens that , so this is not a counterexample, in spite of the fact that the “horizontal” or “vertical” subgroup is not a canonical choice for the diagonal subgroup.
What is a good name for groups with this property? It’s not completely trivial: cyclic groups, for instance, have this property–not that I think this property is important, but names can be amusing…
I have examples of groups and with not (abstractly!) a product of subgroups of . My challenge to you is to find some explicit examples of and prove that doesn’t decompose.
In the end, I think this is a fun problem for a group theory final exam; I think it nicely highlights the difference between “being isomorphic” and “being equal,” though if one completes the challenge as stated, one probably already understands that distinction… So maybe the best reason for blogging about this is that chiastic title.
Constructing a Lie group from a Lie algebra.
November 30, 2006 mathematics
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comes from a connected, simply-connected Lie group . I hadn’t known the proof of this result (and apparently it is rather uglier than one might hope), but  gives a short proof of it, which I presented to the undergraduates in my Lie group seminar. I’ll sketch the proof now.
Theorem. For every Lie algebra , there is a simply-connected, connected Lie group having as its Lie algebra.
First, if , then the exponential map gives , and we define . It turns out is a Lie group, and is its Lie algebra.
If has no center, then is injective, so we have realized as a Lie subalgebra of endomorphisms of a vector space, and by the above, there is a Lie group with as its Lie algebra. Taking its universal cover proves the theorem in this case.
Now we induct on the dimension of the center . Let be a one-dimensional central subspace of , and construct a short exact sequence . But this central extension of by corresponds to a 2-cocycle .
Lemma. Let be the map which differentiates a (smooth!) -cocycle of the group cohomology of . The map is injective.
Consequently, we can find with . Since , by induction there is a Lie group having as its Lie algebra. We build the central extension of by using the cocycle , namely, , where and the operation is . Since , it turns out that the Lie algebra corresponding to is . We finish the proof by taking the universal cover .
 V.V. Gorbatsevich, Construction of a simply connected group with a given lie algebra, Uspekhi Mat. Nauk. 41 (1986) 177–178.
Coxeter group visualization.
November 28, 2006 mathematics
Jenn is a fabulous program for visualizing the Cayley graphs of finite Coxeter groups. The pictures are absolutely beautiful (oh, symmetry!).
Efficient construction of the reals.
October 20, 2006 mathematics
Today in Geometry/Topology seminar, quasihomomorphisms were discussed, i.e., the set of maps such that is uniformly bounded, modulo the relation of being a bounded distance apart. These come up when defining rotation and translation numbers, for instance.
Anyway, Uri Bader mentioned that these quasihomomorphisms form a field, isomorphic to , under pointwise addition and composition. I hadn’t realized that this is a general construction. Given a finitely generated group (with fixed generating set, so we have the word metric on the group), I can define a quasihomomorphism by demanding be uniformly bounded, and where two quasihomomorphisms are equivalent if is uniformly bounded. Let’s call the resulting object for now.
What can be said about ? For instance, what is ?
Outer Space. And real aliens.
October 10, 2006 mathematics
Here is a ridiculously simple question I have wondered about: given , say with , how can I tell if and are conjugate? I suspect I’m being stupid here.
In light of my recent comments on LINCOS and communicating with extraterristrials, I found the article .
Putnam also makes use of the idea of mathematicians from other planets, to more philosophical ends.
 M. Bestvina, The topology of , in: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002: pp. 373–384.
 K. Vogtmann, Automorphisms of free groups and outer space, in: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002: pp. 1–31. https://doi.org/10.1023/A:1020973910646.
 D. Ruelle, Conversations on mathematics with a visitor from outer space, in: Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000: pp. 251–259.
Euler characteristic of closed hyperbolic 4-manifolds.
September 22, 2006 mathematics
By the Gauss-Bonnet theorem, the volume of a hyperbolic 4-manifold is proportional to its Euler characteristic. There are examples, constructed explicitly in  of hyperbolic 4-manifolds with every positive integer as their Euler characteristic. These examples are non-compact (with five or six cusps, I believe). But  observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in  that the Pontrjagin numbers of a hyperbolic manifold vanish. But the signature is a rational linear combination of those Pontrjagin numbers, so . And by Poincare duality, , so is even. A natural question to ask is: does there exist a hyperbolic 4-manifold with ? Now if such an also had , we would know the volume spectrum of closed hyperbolic 4-manifolds.
This certainly seems to parallel the case for 2-manifolds: all negative integers are the Euler characteristic of a hyperbolic 2-manifold, and all even negative integers are the Euler characteristic of a closed hyperbolic 2-manifold.
The vanishing of Pontrjagin numbers for hyperbolic manifolds also holds for pinched negative curvature under some conditions .
It is also a fact that the Stiefel-Whitney numbers vanish for a closed hyperbolic manifold (and the vanishing of the top Stiefel-Whitney class is the same thing as having even Euler characteristic).
 J.G. Ratcliffe, S.T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000) 101–125. http://projecteuclid.org/getRecord?id=euclid.em/1046889595.
 J.G. Ratcliffe, The geometry of hyperbolic manifolds of dimension at least 4, in: Non-Euclidean Geometries, Springer, New York, 2006: pp. 269–286.
 S.-s. Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg. 20 (1955) 117–126.
September 14, 2006 mathematics
Margulis’ amazing arithmeticity theorem says that irreducible lattices in Lie groups of high () rank are arithmetic. But has rank 1, so a question is how to produce non-arithmetic lattices. For , there are non-arithmetic lattices coming from hyperbolic knot complements.
G–P-S  produces higher dimensional examples by taking two hyperbolic (arithmetic) manifolds, cutting along totally geodesic hypersurfaces, and gluing. Are there are examples of non-arithmetic hyperbolic manifolds without any totally geodesics hypersurfaces?
There are complements of ’s in which are hyperbolic, and maybe these would provide some examples.
 M. Gromov, I. Piatetski-Shapiro, Nonarithmetic groups in lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. (1988) 93–103.
August 16, 2006 mathematics
First of all, it should be remarked that according to the Tannakian phylosophy, one can reconstruct a group from the category of its finite-dimensional representations, equipped with the structure of the tensor product.
I suppose one should think of this as the categorification of Pontrjagin duality?
For a long while, I had wondered how this goes; this Introduction to Tannaka Duality and Quantum Groups will probably answer my questions.
Finite subgroups of rotation groups.
April 5, 2006 mathematics
Here is a question that I haven’t been able to find very much about:
What are the finite subgroups of the rotation groups ?
For examples, I can take a Coxeter group, and choose elements corresponding to rotations (e.g., the subgroup generated by products of generators), but that’s not going to produce very many examples.
Orientable 3-manifolds are parallelizable
March 5, 2006 mathematics
Here’s a very easy theorem.
Theorem. All closed orientable 3-manifolds are parallelizable. All closed orientable 3-manifolds are the boundary of a 4-manifold.
Proof. Let be an orientable -manifold. Recall that the Wu class is the unique cohomology class such that , and Wu’s theorem says that . The up-shot is that Stiefel-Whitney classes are homotopy invariants, even though they are defined using the tangent bundle.
Since is orientable, we have . Since , the Steenrod squares and kill everything, so and . By Wu’s theorem, , and . In other words, all the Stiefel-Whitney classes vanish.
Orientability matters; after all, being orientable is the same thing as vanishing. For example, is not parallelizable, since .