Abstract- We describe an algorithm to determine the fundamental crossed module of a handle decomposition of the complement of an embedded surface in $S4$ (constructed from a hyperbolic splitting of it). This thus yields a geometric method to determine the algebraic 2-type of the complement of a knotted surface in 4-space.

References:

Lomonaco, S. J., Jr. The homotopy groups of knots. I. How to compute the algebraic $2$-type. Pacific J. Math. 95 (1981), no. 2, 349--390.

Faria Martins J. The fundamental crossed module of the complement of a knotted surface, arXiv:0801.3921v1 [math.GT], to appear in Trans AMS.