  We will strengthen the known upper and lower bounds on the delta-crossing
number of knots in therms of the triple-crossing number. The latter bound turns
out to be strong enough to obtain (unknown values of) triple-crossing numbers
for a few knots. We also prove that we can always find at least one tangle from
the set of four tangles, in any triple-crossing projections of any non-trivial
knot or non-split link. In the last section, we enumerate and generate tables
of minimal delta-diagrams for all prime knots up to the delta-crossing number
equal to four. We also give a concise survey about known inequalities between
integer-valued classical knot invariants.
