Number theory in low-dimensional topology

Number of Closed Essential Surfaces in Montesinos Knots

We discuss the main theorem, in a recent paper, concerning the number of closed, connected, essential, orientable surfaces of a fixed genus g, up to isotopy, in the complement of a hyperbolic Montesinos knot with 4 rational tangles. This is the first infinite family of 3-manifolds where such counts are precisely known, but simply not zero for all large g. In part 1, we state the main theorem, related results, and background definitions. In part 2, we give a rough sketch of the proof by stating important results and tools that are used.

Paper: The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles 

Bias in the distribution of holonomy on compact hyperbolic 3-manifolds

The study of hyperbolic 3-manifolds draws deep connections between number theory, geometry, and topology. An important geometric invariant of a hyperbolic 3-manifold is the set of its closed geodesics, which are parametrized by their length and holonomy, which measures the “twist” in the geodesic. It turns out that for geodesics of increasing lengths, holonomy is equidistributed throughout the circle. In the first talk, I’ll introduce hyperbolic 3-manifolds and present results on the distribution of holonomy, including equidistribution and bias for compact 3-manifolds. In the second talk, I’ll give background on Selberg’s trace formula, including a non-spherical trace formula, and discuss how this spectral to geometric correspondence is used to prove bias in the distribution of holonomy.

Paper: Bias in the distribution of holonomy on compact hyperbolic 3-manifolds 

Right-angled links in thickened surfaces

In this talk, we will consider links which are alternating on a surface of genus at least one, $S_g$, in the thickened surface $S_g \times I$. We will define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, equivalent to each checkerboard surface consisting of one type of polygon, and equivalent to a set of restrictions on the link's alternating projection diagram. In the traditional setting of alternating knots on S^2 in S^3, the are no knots with this property. We will show that RGCR knots do exist in thickened surfaces. 

Paper: Right-angled links in thickened surfaces  

Algebras of the Whitehead link

Let Γ be the fundamental group of a knot or link complement M. The discrete faithful representation into PSL_2(C) has an associated quaternion algebra. We can zoom out and look at the canonical component E(M) of the character variety X(M). For almost every point on E(M), there is an associated quaternion algebra whose definition is polynomial in the coordinates of the point. We'll see how this algebra behaves for the Whitehead link complement and how the algebra can descend to quaternion algebras of the (m,n)-surgeries thereon.

Ramification for surgeries on knots

Associated to an orientable, finite volume hyperbolic 3-orbifold is a number field and a quaternion algebra over that field. Such quaternion algebras are classified up to isomorphism by their ramification sets, which is a finite set of prime ideals and embeddings into the real numbers. Chinburg, Reid, and Stover show that orbifolds obtained by Dehn surgery on knots whose Alexander polynomial satisfies some condition have quaternion algebras ramifying above a finite number of rational primes. In my thesis I give examples where the corresponding set is infinite for knots that fail this condition.

Papers: 

1. A conjecture of Chinburg-Reid-Stover for surgeries on twist knots 

2. Arithmetic of the canonical component of the Knot 7_4