Patterns in Knot Floer Homology

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 $\log r(K) < a \cdot Vol(K)$ for all knots $K$ where $r(K)$ is the total rank of knot Floer homology (KFH) of $K$ and $Vol(K)$ is the hyperbolic volume of $K$. (2) 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. (3) There exist constants $a, b \in R$ such that $\log \det(K) < a \cdot Vol(K) + b$ for all knots $K$ where $det(K)$ is the knot determinant of $K$.Comment: The dataset is available on Zenodo (doi.org/10.5281/zenodo.7879466). The code is available on GitHub (github.com/eivshina/ patterns-in-knot-floer-homology