Invariants of high-dimensional knots

Question

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?

Answer 1

There are many results in this direction, dating mostly from the mid-70's to early 80's. All require some degree of connectivity of the complement, meaning that $\pi_i(S^{n+2} -K) = \pi_i(S^1)$ for $i\leq m$ for some choice of $m$. Levine's results cited above are the setting where $n$ is odd, and $m=(n+1)/2$. Kearton (An algebraic classification of some even-dimensional knots. Topology 15 (1976), no. 4, 363–373.) had analogous results in even dimensions.

The most systematic treatment of these problems was in a series of papers by Farber, (Isotopy types of knots of codimension two. Trans. Amer. Math. Soc. 261 (1980), no. 1, 185–209, and An algebraic classification of some even-dimensional spherical knots. I, II. Trans. Amer. Math. Soc. 281 (1984), no. 2, 507–527, 529–570.) where he treats the case when $m$ is roughly $n/3$. The classification is really a translation of knot theory into problems of stable homotopy theory; the basic object of study is a homotopy-theoretic generalization of the Seifert pairing. In special cases, one can get actual algebraic invariants for the classification.

For codimension-two embeddings of other kinds of manifolds, much less is known. There is a large literature on embedded surfaces in the 4-sphere, with some classification results in the topological setting. In higher dimensions, the starting point might be the comprehensive work of Cappell and Shaneson (The codimension two placement problem and homology equivalent manifolds. Ann. of Math. (2) 99 (1974), 277–348.)

Answer 2

One result is due to Jerry Levine (see Levine, J. An algebraic classification of some knots of codimension two. Comment. Math. Helv. 45 1970 185–198.)

He showed that if $K\cong S^{2k-1}$ is simple knot in $S^{2k+1}$ with $k>1$, then it is determined up to isotopy by the Seifert matrix. Here a knot is called simple if the homotopy groups of the knot complement below the middle dimension are the homotopy groups of the complement of a trivial knot, i.e. if $\pi_i(S^{2k+1}\setminus K)\cong \pi_i(S^1)$ for $i=1,\dots,k-1$.

Answer 3

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.