Asymptotics for the number of Simple $(4a+1)$-Knots of Genus 1

TL;DR

This paper studies the asymptotic count of certain simple knots with specific Alexander polynomials, linking algebraic forms to number theory heuristics, and provides bounds on their growth rate.

Contribution

It establishes a connection between knot counting and quadratic forms, applies Cohen-Lenstra heuristics, and proves bounds on the number of such knots related to prime parameters.

Findings

Asymptotic count of knots is conjectured to be proportional to $X^{3/2}/ ext{log} X$.

The contribution from prime-related parameters is bounded by $O(X^{3/2}/ ext{log} X$).

Total number of knots grows slower than $X^{3/2}$, i.e., is $o(X^{3/2})$.

Abstract

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of $\mathbb{Z}[1/m]$-equivalence classes of binary quadratic forms of discriminant $1-4m$, for $m$ running through the same range. Our heuristics, based on the Cohen-Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$, and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ prime is bounded above by $O(X^{3/2}/\log X)$, and that the total itself is $o(X^{3/2})$.