Foliations and genera of links

Abstract

IN THIS paper we define a simple geometric operation, disc decomposition, which when successfully applied to a surface R in S3 implies that R is a surface of minimal genus for the oriented link L = dR, that there exists a C”, transversely oriented foliation 9 of S3 -&T(L) such that 9 I$ aN(L), R is the unique compact leaf (when R # D*), the union of the noncompact leaves fibre over S’ with fibre a connected non-compact leaf and if L is a knot, longitudinal (O-frame) surgery yields an irreducible 3-manifold. If L is a non-split oriented alternating link, link of I 9 crossings, knot of I 10 crossings, flat arborescent link (these include the pretzel knots), or fibred link then L bounds a surface which is disc decomposable. We show that the surface gotten by applying Seifert’s algorithm to an alternating projection is disc decomposable thereby giving a geometric proof of the result due independently by Murasugi [123 and Crowell[3] that such a surface has minimal genus. We give tables indicating the genera of the oriented nonalternating knots and links of < 10 and < 9 crossings respectively. The computation of the genera of the arborescent links including the disc decomposability of surfaces bounding flat ones and the decomposability of the fibres of fibred links can be found in [S] and [7]. The non-splitness condition is necessary by Novikov’s work [13].