Counting simple knots via arithmetic invariants

Abstract

Knot theory and arithmetic invariant theory are two fields of mathematics that rely on algebraic invariants. We investigate the connections between the two, and give a framework for addressing asymptotic counting questions relating to knots and knot invariants.

We study invariants of simple $(2q-1)$-knots when $q$ is odd; these include the Alexander module and Blanchfield pairing. In the case that $q=1$, simple $1$-knots are exactly knots as classically defined. In the high-dimensional cases of $q \ge 3$, the theory is different, and simple knots are exactly classified by these algebraic invariants.

These invariants connect to arithmetic invariant theory by way of the theory of Seifert matrices, which are related to the $\Z$-orbits of the adjoint representation $\Sym^2(2g)$ of the algebraic group $\Sp_{2g}$. We classify the orbits of this representation over general fields and over $\Z$. These techniques are modeled after those of Bhargava, Gross, and Wood in arithmetic invariant theory, but also have much in common with methods used by Trotter and others in the topological context. We explain how our results fit into this topological context of the Alexander module, Blanchfield pairing, and related invariants.

In the final section, we look at how this connection can be used to asymptotically count simple knots and Seifert hypersurfaces ordered by the size of their Alexander polynomial. For knots of genus $1$, the theory of binary quadratic forms yields an explicit count for Seifert surfaces. We also conjecture heuristics for the asymptotic number of genus $1$ knots. These heuristics imply that most such knots have Alexander polynomial of the form $p t^2 + (1-2p) t + p$ where $p$ is a positive prime number. Using sieve methods, we obtain an upper bound for the asymptotic count of such knots that agrees with our heuristics.