ABSTRACT
C lassification of knot projections under RI and RII was obtained in the latter half of the 20th century ([8], 1997). Originally, Khovanov obtained
the classification by considering doodles. The notion of doodles was introduced by Fenn and Taylor ([2], 1979). Extending the notion of doodles, Khovanov showed the existence and uniqueness of the generators of knot projections on a plane. This chapter discuss Khovanov’s classification theorem for knot projections on a sphere in the manner described in Ito-Takimura ([4], 2013). In this chapter, knot projections imply knot projections on a sphere unless specified otherwise. Recently, a classification theorem under RI and strong RII (resp. weak RII) was obtained by Ito-Takimura [5] by extending their theorem developed in 2013 [4]. Ito-Takimura [5] also obtained a new additive integervalued invariant |τ(P )| of a knot projection P , called circle number , under RI and strong RII. The other known additive integer-valued invariant a(P ) of a knot projection under RI and strong RII was obtained by Arnold ([1], 1994), but |τ(P )| and a(P ) are independent of each other. The other additive integer-valued invariants under RI and strong RII are unknown.
