Patterns in Knot Floer Homology

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 ∈ ℝ_{>0} such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number c → ∞: log r(K) < a · 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 ∈ ℝ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number c → ∞: log det(K) < a · 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) < L/(1 + exp(−k · (x − x₀))) + b where L, x₀, k, b are constants.