Edited 2/9 after discussion with Dylan Thurston
It seems unlikely that the obvious knot projections can be simplified by more than a constant factor, so a quadratic lower bound for the expected minimum crossing number seems likely. Crossing number by itself though is a strange measure of complexity, and it is hard to compute. However, it's bounded below by hyperbolic volume of the knot complement.
It would be possible to get some experimental evidence by feeding output of your random
process through snappea, and looking at the distribution of hyperoblic hyperbolic volume. However, I think hyperbolic volume probably grows at a less than quadratic rate. You can imagine thickening the knot into a growing solid torus, pushing outward until every part of the
boundary has bumped into other boundary --- similarly to a Voronoi subdivision. I think
the expectation for the number With
tubes of faces grows slower than diameter some constant times $n^2$. This number n^{-.5})$, the total volume of tubes is an
upper bound for hyperbolic on the
order of the volume (assuming of the topology doesn't get too crazy).
Alternativelyball, you could look for so typical tube spacing should be $O(n^{-.5})$.
This suggests the number of faces in this subdivision should be $O(n^{3/2})$, which
would give a triangulation (not necessarily linear) having $O(n^{3/2})$ tetrahedron where the knot is on in the
1-skeleton--- , implying that the typical Gromov norm or hyperbolic volume is linearly bounded by
probably
grows as $O(n^{3/2})$. This would only imply $n^{3/2}$ crossings.
Marc Lackenby, in SPECTRAL GEOMETRY, LINK COMPLEMENTS AND SURGERY DIAGRAMS, developed a beautiful method to give lower bounds for crossing numbers for knots. His method possibly could be applicable to improve this situation, provided the number of tetrahedraCheeger constants for these manifolds can be shown to be not too small.
It's also possible that one could estimate the degree of the Alexander polynomial, to get an estimate of the crossing number.
