Dehn-surgery along a torus T 2-knot
A \(T^ 2\)-knot means an embedded 2-torus. The author defines a torus \(T^ 2\)-knot, a generalization of a classical torus knot, as an incompressible embedded 2-torus in the boundary of a tubular neighborhood of an unknotted 2-torus in the 4-sphere, where an unknotted 2-torus means an embedded torus which bounds \(S^ 1\times D^ 2\) in \(S^ 4\). A torus \(T^ 2\)-knot turns out to be an untwisted or twisted spin of a classical torus knot. Dehn surgery means an operation of cutting off the tubular neighborhood of a knot and pasting it back via a diffeomorphism between the boundaries. The author shows that a 4-manifold obtained by Dehn surgery along a torus \(T^ 2\)-knot is (i) the total space of a good torus fibration over \(S^ 2\) or (ii) \(L_ m\) or \(L_ m'\), the manifolds obtained by Dehn surgeries along an unknotted 2-torus, or (iii) an irrational connected sum along circles of either \(L_ m\) or \(L_ m'\) and \(L(n,r)\times S^ 1\). The author also shows that a 4-manifold obtained by Gluck surgery is also obtained by Dehn surgery along some \(T^ 2\)-knot in \(S^ 4\).
scientific article; zbMATH DE number 4024309