Arithmetic of Dehn surgery points
Joint work with Ted Chinburg and Matthew Stover.
1. Invariants of hyperbolic 3 -manifolds
The key players are finite volume hyperbolic 3-manifolds. An example is the figure-eight complement, whose fundamental group is generated by two matrices
Remove a neighborhood of the end, the boundary is a 2-torus. Glue a solid torus instead (there are several ways, indexed by pairs of integers ). Thurston that the resulting closed manifold is hyperbolic for all but finitely many .
Associated to fundamental group are
- A trace field . Infinitesimal rigidity implies that is a number field.
- A quaternion algebra , where is the holonomy. It turns out to be 4-dimensional.
Note that is used to produce an example of a full tower of finite Galois covers of a hyperbolic manifold which are homology spheres.
The snap software (a merge of snappy and pari) is convenient to design approximations and deformations of hyperbolic 3-manifolds. Experiments indicate that interesting ramification behaviour occurs among trace fields and quaternion algebras of manifolds obtained by Dehn surgery.
Theorem 1 Let be a hyperbolic knot. Suppose that its Alexander polynomial satisfies
(*) Square roots of roots of generate the same field over as .
Let be the quaternion algebra of the Dehn surgery manifold . Then there exists a finite set of primes such that every prime over over which some ramifies divides an element of .
3. Arithmetic of the canonical curve
Thurston showed that in the character variety of a 1-cusped hyperbolic manifold group, there is canonical complex curve, which contains all Dehn surgeries. For the figure eight knot, it is an elliptic curve. For the -Pretzel knot, it is a rational curve.
Kronheimer and Mrowcka proved that every nontrivial knot has an irreducible -representation. Reducible representations are also of interest. Thanks to a not so well known theorem by de Rham, they are related to roots of the Alexander polynomial.
Over the function field of the canonical curve, there is a quaternion algebra. Azumaya algebras generalize central simple algebras over fields, where fields are replaced with commutative local rings. They captures the local behaviour at a point of the canonical curve. It does not always exist. The criterion for its existence (Azumaya positivity) turns out to be (*).
The fact that Azumaya positivity implies the ramification property in the theorem is a theorem by Harari.
4. Link to Heegard-Floer homology?
Conjecture. If is a hyperbolic -space knot, it is never Azumaya positive.
-space is defined in terms of Heegard-Floer homology.