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 aβR>0β such that logr(K)<aβ 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)<1+exp(βkβ (xβx0β))Lβ+b where L,x0β,k,b are
constants.
(3) There exist constants a,bβR such that logdet(K)<aβ 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