The Jones Polynomial, Genus and Weak Genus of a Knot

In his book [Ad, p. 105 bottom], C. Adams mentions a result of Morton that there exist knots, whose genus g is strictly less than their weak genus g̃, the minimal genus of (the surface of Seifert’s algorithm applied on) all their diagrams. This observation appears just as a remark in [Mo], but was very striking to the author. Motivated by Morton’s example, the author started in a series of papers [St2, St, St3] the study of the invariant g̃. A key role in what we can say so far about g̃ plays [St2, theorem 3.1], saying that knots of given g̃ decompose into finitely many sequences of the kind introduced in [St4], and called there “braiding sequences”, that is, can be obtained from finitely many diagrams by successive applications of antiparallel twists at a crossing ! : (1)