ABSTRACT
H alf-twisted splice is the inverse of a splice of a double point of a knot projection for obtaining a knot projection from another knot projection.
A “splice” is a replacement of a sufficiently small disk with another disk while preserving its boundary on a knot projection; here, a local part of two paths at a double point is replaced with two sub-curves with no double points. Splice is considered to be the basic tool in knot theory. There are two types of splices, one is called a Seifert splice and the other is the inverse of a half-twisted splice, denoted by A−1 [1]. In this study, we suppose that every knot projection has at least one double point. For a given knot projection, the notion of the inverse
of the half-twisted splice A−1 induces Shimizu’s reductivity r(P ) that is the minimum number of applications of A−1, recursively, to obtain a reducible knot projection from P . The reductivity is closely related to an open question concerned with an unavoidable set. Roughly speaking, an unavoidable set is a set for which the following condition holds: any knot projection must contain a part that is sphere isotopic to one of the elements. If a knot projection contains a single 3-gon, the types of 3-gons can be easily classified into four types. Two of them that appear on two sides of a weak RIII are denoted by A and B; the other two that appear on the two sides of a strong RIII are denoted by C and D. The open question is that for a reduced knot projection, is {2-gon, 3-gons of type A, B, and C} an unavoidable set? This open question concerned with an unavoidable set is also related to the triple chord theorem [3], which is stated as follows: the chord diagram of any prime knot projection with no 1-gons and 2-gons contains triple chords.
