Patterns in Knot Floer Homology

Ekaterina S. Ivshina

Abstract:

Based on the data of 12-17-crossing knots, we establish three new conjectures about the hyperbolic volume and knot cohomology:

(1) There exists a constant $a\in {R}_{>0}$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \to \infty$:

$$ \log r(K) < a \cdot \mbox{Vol}(K) $$
for a knot $K$ where $r(K)$ is the total rank of knot Floer homology (KFH) of $K$ and $\mbox{Vol}(K)$ is the hyperbolic volume of $K$.

(2) There exist constants $a,b\in {R}$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \to \infty$:

$$ \log \mbox{det}(K) < a \cdot \mbox{Vol}(K) +b$$

for a knot $K$ where $\mbox{det}(K) $ is the knot determinant of $K$.

(3) Fix a small cut-off value $d$ of the total rank of KFH and let $f(x)$ be defined as the fraction of knots whose total rank of knot Floer homology is less than $d$ among the knots whose hyperbolic volume is less than $x$. Then for sufficiently large crossing numbers, the following inequality holds

$$f(x) < \frac{L}{1 + \exp(-k\cdot(x-x_0))} + b$$
where $L ,x_0, k, b$ are constants.