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A classic result in 3-manifold topology is Neuwirth's conjecture, which states that the fundamental group of a knot complement is a free product of two proper subgroups amalgamated along a free group. This was proven by Culler and Shalen using the algebraic geometry of representation varieties of 3-manifold groups into $SL_2 C$. Since this is an affine variety, one may associate at least two ideal points (the non-triviality of the representation variety follows from Thurston's geometrization theorem). Associated to these ideal points is an action on a Bass-Serre tree, and then a technique of Stallings associates to this a separating (for at least one ideal point) surface with boundary, and the desired amalgamated product.