Sigmoid density conjecture for low-rank knot Floer homology

Fix a small cut-off value $d$ for the total rank of knot Floer homology. For $x$ a volume threshold, let $f(x)$ be the fraction of knots whose total knot Floer homology rank is less than $d$ among knots whose hyperbolic volume is less than $x$. Sigmoid density conjecture. For sufficiently large crossing numbers, there are constants $L$, $x_0$, $k$, and $b$ such that

$$f(x)<\frac{L}{1+\exp\bigl(-k\cdot(x-x_0)\bigr)}+b.$$

The conjecture is motivated by sigmoid-shaped fits to data for non-alternating knots with 12–17 crossings; the source does not provide a resolution, and the quantification of “sufficiently large crossing numbers” remains open.