Patterns in Knot Floer Homology

TL;DR

This paper investigates relationships between knot Floer homology, hyperbolic volume, and knot determinants in knots with 12-17 crossings, proposing three new conjectures supported by data analysis.

Contribution

It introduces three novel conjectures linking knot Floer homology and hyperbolic volume based on empirical data from knots with 12-17 crossings.

Findings

Conjecture that the total rank of knot Floer homology grows exponentially with hyperbolic volume.

Conjecture relating knot determinant to hyperbolic volume with linear bounds.

Empirical evidence for a sigmoid-like distribution of knot Floer homology ranks relative to volume.

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 $c \to \infty$: $\log r(K) < a \cdot Vol(K)$ for a knot $K$ where $r(K)$ is the total rank of knot Floer homology (KFH) of $K$ and $Vol(K)$ is the hyperbolic volume of $K$. (2) 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 $c \to \infty$: $\log \det(K) < a \cdot Vol(K) + b$ for a knot $K$ where $\det(K)$ 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.