Deep and shallow slice knots in 4-manifolds

We consider slice disks for knots in the boundary of a smooth compact 4-manifold X 4 X^{4} . call knot K subset-of partial-differential upper X"> K ⊂ ∂ encoding="application/x-tex">K \subset \partial X deep slice encoding="application/x-tex">X if there is properly embedded alttext="2"> 2 encoding="application/x-tex">2 -disk with K"> encoding="application/x-tex">K , but not concordant to unknot collar neighborhood alttext="partial-differential times I"> ×<!-- × <mml:mi>I encoding="application/x-tex">\partial \times {I} boundary. point out how this concept relates various well-known conjectures and give some criteria nonexistence such deep knots. Then we show, using Wall self-intersection invariant result Rohlin, that every consisting just one 0- nonzero number 2-handles always has end by considering 4-manifolds where bounds an disk interior. A generalization Murasugi-Tristram inequality used show does exist compact, oriented alttext="4"> encoding="application/x-tex">4 -manifold V"> V encoding="application/x-tex">V spherical S cubed equals S 3 = {S}^{3} = V via null-homologous disk.