Cochran Invariants

Recently Tim Cochran passed away unexpectedly at the age of 59. Everyone who knew him was shocked and saddened at his untimely death, and my deepest condolences extend to all who were affected by this. Coincidentally, I was thinking about a cool paper he wrote in 1984 where he introduced a method he would later explore in much greater depth in an AMS Memoir book, the method of "derived links."

Suppose you have a link of two components with linking number $0$. Each component spans a Seifert surface, and the two surfaces intersect generically along a $1$-manifold. It is not hard to show that you can arrange for this $1$-manifold to be connected: so it is a circle. Either of the two surfaces determines an integer framing of this circle. This framing is called the Sato-Levine invariant and is independent of the choice of Seifert surfaces. We will denote this invariant by $\beta^1$. It is a fun exercise to compute this for the Whitehead link. (The answer is $\pm1$ depending on conventions.)

Cochran's idea was to iterate this construction to define a sequence of higher order invariants $\beta^i$ defined as follows. Form a new link (a derived link) from the original link $L$ by tossing out the second component and replacing it with the intersection curve of the two surfaces. We will call this link $D(L)$.  Now we define $\beta^2(L)=\beta^1(D(L))$, and it turns out this is well-defined. In general $D^k(L)$ is defined by keeping the first component fixed and replacing the second component by intersection of the seifert surfaces of $D^{k-1}(L)$. Then $\beta^k(L)=\beta^1(D^{k-1}(L))$. The amazing thing is that this is a well-defined integer.

Cochran proves that $\beta^i(L)$ are integer lifts of Milnor's $\mu$ invariants $\mu_{1122...22}$, where there are $2i$ "2"s. This is interesting because normally $\mu$ invariants are only well-defined modulo the gcd of lower order invariants, and it also gives a cool topological interpretation of them. Cochran would later go on to show that one could define all $\mu$ invariants in this way using derived links, but one cannot in general remove the indeterminacy as in this case.

Cochran also proves that these $\beta^i(L)$ are essentially equivalent to a polynomial invariant called the $\eta$ function, due to Kojima.  Briefly, let $L=(M,K)$ be a two component link as above, and consider the infinite cyclic cover of the complement, $Y$, and let $\Delta(t)$ be the symmetrized Alexander polynomial of the second component $K$. Let $z$ be a lift of $M$ to $Y$, $z_0$ a nearby lift of an untwisted parallel of $K$ and $t_*$ a generator of the covering transformation group. Then $A(t_*)$ kills $z\in H_1(Y)$ and $A(t_*)(z)=\partial d$ for some 2-chain $d$ in $Y$. Then Kojima's $\eta$ function is defined as $$\eta(L)=\sum_{n=-\infty}^\infty\frac{1}{A(t)}(z_0\cdot t_*^nd)t^n.$$ Cochran proves that under the change of variables $x=(1-t)(1-t^{-1})$,  $$\eta(L)=\sum_{i=1}^\infty \beta^i(L)x^i.$$ This brings me to why I was interested in this.  My collaborators Rob Schneiderman and Peter Teichner were studying the Kirk invariant of immersed $2$-spheres in the $4$-sphere, which turns out to be  related to the Kojima invariant in the following way. Take a link of $2$ components with linking number $0$ and both components unknotted. Think of this link as lying on a copy of $S^3$ in $S^4$. Cap off the first component into the upper hemisphere of $S^4$ with a standard disk, and homotope the second component to a trivial knot in the complement of the first component, also rising up into the upper hemisphere. Do the same thing into the lower hemisphere but with components reversed. This gives two disjoint self-intersecting spheres in the $4$-ball. It turns out that the Kirk invariant of this link of $2$-spheres is the $\eta$ invariant of the classical link that we started with! This was noted by Kirk in his original paper.

Peter and Rob were at that point unaware of the connection to Cochran's work, and had independently defined the $\eta$ invariant in this simple case. They conjectured that it was related to well-known invariants like $\mu$ invariants. After discussing this with them, I realized that the only possible $\mu$-invariants that this could be related to were the $\mu_{1122...22}$ invariants, and that a basis for $2$-component links  of the above form modulo equivalence of their $\eta$ invariants was given by clasper surgeries on the unlink by trees $t_k$ defined as follows. Take a line segment with the two endpoints labeled by "1" and attach  $k$ hairs in a row labeled by "2." I was puzzled by the fact that after calculating $\mu_{112...2}$ on my tree basis, I was getting
$$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\mu_{1122}      &   1& 2& 0& 0& 0& 0&\\
\mu_{112222}     &*&  *& 1 &2& 0& 0&\\
\mu_{11222222} &* &* &*& * & 1 &2\\
\end{array}
$$
The *'s indicate places where $\mu$ invariants are not defined due to their indeterminacy. In my head, I was thinking of this table (incorrectly) like this:
 $$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\mu_{1122}      &   1& 2& 0& 0& 0& 0&\\
\mu_{112222}     &0&  0& 1 &2& 0& 0&\\
\mu_{11222222} &0 &0 &0& 0 & 1 &2\\
\end{array}
$$
This indicated that the $\mu$ invariants were not a dual basis, since this matrix was not diagonalizable and in fact half of the columns were lacking pivots. So there were some other mysterious invariants floating around, but what the heck could they be? In a sense this is indeed true, there are other mysterious invariants floating around! Namely, Cochran's $\beta^i$ invariants, which are integer lifts of the $\mu$ invariants. In fact, one can fill in the chart above with the $\beta^i$'s as follows:
 $$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\beta^1     &   1& 2& 0& 0& 0& 0&\\
\beta^2    &0&  1& 1 &2& 0& 0&\\
\beta^3 &0 &0 &-3& -4 & 1 &2\\
\end{array}
$$
which is perfectly consistent with the $\beta^i$'s forming a complete invariant system. Cochran had already figured out the problem we were studying 30 years ago.