Delta-crossing number for knots

A Delta-crossing tangle is a tangle of three arcs with three crossings, which appears in a Delta move (or Delta unknotting operation). A Delta-crossing diagram is a diagram which can be decomposed into Delta-crossing tangles joined by simple arcs. We prove that every knot has a Delta-crossing diagram, and then investigate the Delta-crossing number which is the minimum number of Delta-crossing tangles among all Delta-crossing diagrams of the given knot. We obtain upper and lower bounds on the number in terms of the ordinal crossing number and genus. We also determine the number for prime knots with nine crossings or less except six knots.