ABSTRACT
A fundamental fact related to knot projections is that any two knot projections are related by a finite sequence generated by ∆, RI, RII,
and RIII. The equivalence generated by ∆, i.e., the equivalence under a finite sequence generated by ∆, is called sphere isotopy (or plane isotopy when plane curves are considered). The equivalence relation generated by ∆, RII, and RIII is known as regular homotopy (i.e., RI is forbidden). The classification problem for the equivalence relation generated by ∆, RI, and RII (i.e., RIII is forbidden) was solved by Khovanov ([2], 1997). The case for which RII cannot be used is still unexplained; moreover, the knot projections that can be trivialized have not been determined yet!
