ABSTRACT
F or refined Reidemeister moves, i.e., RI, strong RII, weak RII, strong RIII, and weak RIII, we have been considering equivalence relations, each
of which is generated by two refined Reidemeister moves of a tuple of type (RI, y): (RI, ∅), (RI, strong RII), (RI,weak RII), (RI, strong RIII), and (RI,weak RIII). For each equivalence relation, we have already determined the equivalence class that contains a trivial knot projection. We shall take one step further and consider the equivalence relations generated by three refined Reidemeister moves of type (RI, x, y): (RI, strong RII, strong RIII), (RI, strong RII,weak RIII), (RI,weak RII, strong RIII), (RI,weak RII,weak RIII). As shown by Ito-Takimura [3], all knot projections are related to a trivial knot projection under an equivalence relation (RI, x, y), as shown above, except for (RI,weak RII,weak RIII). The equivalence relation corresponding to (RI,weak RII,weak RIII) is called weak (1, 2, 3) homotopy. It is still not known which knot projection is equivalent to a trivial knot projection under weak (1, 2, 3) homotopy. Ito-Takimura [3] introduced the first invariant and showed that there exist infinitely many equivalence classes of knot projections under weak (1, 2, 3) homotopy.
