Wirtinger presentations for higher dimensional manifold knots obtained from diagrams (Q2717749)

Wirtinger presentations allow a method of describing the fundamental group of a classical knot in 3-space and are read off a knot diagram. For oriented surfaces in 4-space, J. S. Carter and M. Saito have developed an analogue method to read Wirtinger relators of the fundamental group of the complement off a diagram. By modifying this method, making it work for any closed smooth or PL submanifold of dimension \(n\) in Euclidean \((n+2)\)-space, \(n\geq 2\), the author finds a way to construct a Wirtinger presentation of the fundamental group of the complement of the submanifold from a regular projection diagram. Local flatness of the submanifold is not required. Moreover, any such group can be realized by the corresponding group of a suitable surface in 4-space.