Isotopy Invariants of 2-manifold

Question

What are the ambient isotopy invariants of a 2-manifold with boundary embedded in $R^3$? Is there a good reference for the case of genus 0?

Answer 1

Well, the first invariant is the isotopy class of the boundary, so there's all of knot and link theory. If you fix the isotopy class of the boundary, then there are knots with inequivalent Seifert surfaces. (Typically, one would look at surfaces of minimal genus with the given boundary.) See older papers by Alford and Eisner (Knots with infinitely many minimal spanning surfaces. Trans. Amer. Math. Soc. 229 (1977), 329–349.) Eisner shows that for non-fibered knots, there is always an infinite number of spanning surfaces, up to isotopy. In the end, the obstruction comes from the fundamental group.

A recent paper of Hedden-Juhasz-Sarkar (On sutured Floer homology and the equivalence of Seifert surfaces, AGT 13 (2013), no. 1, 505–548) gives obstructions to isotopy by combining the Seifert form with sutured Heegaard Floer homology. See also Altman (Sutured Floer homology distinguishes between Seifert surfaces).