TITLE: Invariants of high-dimensional knots
QUESTION [10 upvotes]: In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about the existence of complete or 'almost' complete algebraic invariants for codimension-2 embeddings of arbitrary manifolds $M^n$ into $S^{n+2}$? Are there any special classes for which such complete invariants exist?

REPLY [8 votes]: I feel that I know both too much and too little about high-dimensional knots and links to answer the question properly. In particular, my 1998 book High-dimensional knot theory does not go beyond what was known by Levine in 1970, as far as complete algebraic invariants for codimension-2 embeddings of arbitrary manifolds $M^n$ into $S^{n+2}$ are concerned. My more recent collaboration with Maciej Borodzik and Andras Nemethi does concern invariants of such embeddings, e.g. proving in the preprint Codimension 2 embeddings, algebraic surgery and Seifert forms that the $S$-equivalence classes of Seifert matrices are isotopy invariants just as in the case $M=S^n$ ($n$ odd), but they are not complete invariants. And then there are boundary links, which deserve an answer by themselves.