  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 $o(X^{3/2})$.
