New examples of non-slice, algebraically slice knots (Q2781318)

A link is strongly slice if it bounds a disjoint union of smooth 2-disks in a 4-ball. In particular, a strongly slice knot is simply called a {slice knot}. A knot is called {algebraically slice} if a Seifert form vanishes on some half-dimensional summand of \(H_1(F; {\mathbb Z})\), where \(F\) is a Seifert surface for the knot. Such a summand is called metabolizer for algebraically slice.NEWLINENEWLINENEWLINEIt is known that a slice knot is algebraically slice. \textit{A.J.Casson} and \textit{C.McA.Gordon} [in an article in ''A la recherche de la Topologie perdue'', ed. by Guillou and Marin, Progress in Mathematics, Vol.62 (1986; Zbl 0597.57001) and Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 39-53 (1978; Zbl 0394.57008)] proved that there are algebraically slice knots that are not slice. It is easy to see that if a basis for the metabolizer for an algebraically slice knot \(K\) is represented by a strongly slice link, then \(K\) is slice.NEWLINENEWLINENEWLINEThe following question is asked: If \(K\) is algebraically slice and if a basis of the metaolizer is represented by slice knots, then is \(K\) slice? Clearly, the question is true for genus one knots. The author gives genus two counterexamples. Such an example has already been given by \textit{R.Litherland} [A formula for the Casson-Gordon invariant of a knot, preprint], but the paper never appeared. Recently, \textit{T.D.Cochran}, \textit{K.E.Orr} and \textit{P.Teichner} [Ann. Math. (2) 157, 433-519 (2003; Zbl 1044.57001)] give alternative examples to the ones presented here.