ABSTRACT
T he first break through, i.e., the first example of a non-trivial homotopy equivalence class under RI and RIII, was obtained by Hagge and Yazinski
[1]. They first answered O¨stlund’s question, i.e., they showed that there exists a knot projection such that the knot projection cannot be related to a trivial knot projection by a finite sequence generated by RI and RIII [3]. Surprisingly, they did not use or obtain any invariant to prove their theorem, which is important to note. Although a new topological invariant under a Reidemeister move is often useful to classify knot projections by using the other Reidemeister moves, it is important for us to find a new invariant. On the other hand, proving the impossibility (i.e., proving that there exists a knot projection that cannot be related to a trivial knot projection) without using an invariant may not be easy to understand.
