Upper and lower bounds on delta-crossing number and tabulation of knots up to four delta-crossings

In this paper, we will strengthen the known upper and lower bounds on the delta-crossing number of knots in terms 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 a particular 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.