On the slice genus of knots

Abstract

Given a knot K in the 3-sphere, the genus of K, denoted g(K), is defined to be the minimal genus for a Seifert surface for K. The slice genus gs(K) is defined to be the minimal genus of an oriented surface G admitting a smooth proper embedding in the 4-ball which maps CG to K. If we insist the embedding of G have no local maximum with respect to the radial function, we obtain the ribbon genus gr(K) instead. Thus a knot is slice (ribbon) if and only if gs(K) = 0 (gr(K)= 0). It is clear that g.~(K)<g,.(K)<=g(K). There are well known lower bounds on gs(K) given by invariants of a Seifert matrix for K. These are all included in the invariant re(K) [8] defined by Taylor. It gives the best possible bound based on a Seifert matrix, re(K) vanishes if and only if the Seifert pairing is metabolic. If this is the case, K is called algebraically slice. The work of Casson and Gordon [1, 2, 5] showed that certain algebraically slice knots are not in fact slice. We generalize the main theorem of [1]. As an application, we give a sequence of algebraically slice knots Q, such that gs(Q,)=g(Q,,)=n. We also study the slice genus of K, 41=K, where K, denotes the t twisted double of the unknot. We show for example that g.~(K 12 =~ K 1 z ) = 2 . K I2 is algebraically slice but not slice by [ l ] . Section 1 has some preliminaries on the linking form. In Sect. 2, we state and prove our main theorem. In Sect. 3 we give our examples. In this paper, all manifolds are oriented. We use e to denote the Euler characteristic.