Cobordism of algebraic knots
Any complex isolated hypersurface singularity f: (C\({}^{n+1},0)\to (C,0)\) determines an algebraic knot in \(S^{2n+1}\), the link of the singularity. The author shows that the cobordism type of an algebraic knot depends on the Dynkin diagram with respect to only weakly distinguished bases. More precisely: If there exist weakly distinguished bases for the middle homology of the Milnor fibers of two functions with isolated singularities at the origin such that the Dynkin diagrams summarizing the intersection forms coincide, then the singularities of the functions are cobordant. In particular there exist topological distinct algebraic knots which are cobordant. In order to prove the main result, the author introduces the ``link of a perturbation'' and, as a first step, constructs and analyzes a cobordism between the complement of these links of two singularities with common diagram. Then this cobordism is replaced by another one (which is obtained by surgery modifications) such that the new cobordism relating the link complements is an (integer) homology product. This cobordism is extended to an h-cobordism from \(S^{2n+1}\) to itself which is diffeomorphic by the h-cobordism theorem to \(S^{2n+1}\times I\). In particular the knots are cobordant.
complex isolated hypersurface singularity
bases for the middle homology of the Milnor fibers
scientific article; zbMATH DE number 4096400
scientific article