  We investigate several conjectures in geometric topology by assembling
computer data obtained by studying weaving knots, a doubly infinite family
$W(p,n)$ of examples of hyperbolic knots. In particular, we compute some
important polynomial knot invariants, as well as knot homologies, for the
subclass $W(3,n)$ of this family. We use these knot invariants to conclude that
all knots $W(3,n)$ are fibered knots and provide estimates for some geometric
invariants of these knots. Finally, we study the asymptotics of the ranks of
their Khovanov homology groups. Our investigations provide evidence for our
conjecture that, asymptotically as $n$ grows large, the ranks of Khovanov
homology groups of $W(3,n)$ are normally distributed.
