e \[C1\]). Since $c^+(x,y)$ is rationally null-homologous, there is a surface $V_{c(x,y)}$ whose boundary is “$k$ times $c^+(x,y)$” (in the sense above). We could then define $c(x,x,y)$ to be $V_x\cap V_{c(x,y)}$, an embedded oriented circle on $V_x$. If $c(x,x,y)$ is rationally null-homologous, then $\b^2(x,y)$ is defined as ${\ensuremath{\ell k}}(c(x,x,y),c^+(x,x,y))$ and we may also continue and define $c(x,x,x,y)$. In general $c(x,x\dots,x,y)=c(x^n,y)$ will be able to be defined using the chosen surfaces if $c(x^{n-1},y)$ is defined and is also rationally null-homologous (but to do so involves one more choice of a bounding surface). Once $c(x^{n},y)$ is defined and is rationally null-homologous, we may define $\b^n(x,y)$. In general, we do not claim that the value of $\b^n(x,y)$ is independent of the choices of surfaces. Therefore the invariants can be thought of as taking values in a set, just like Massey products. This indeterminacy will not concern us here, for we are only interested in the first non-vanishing value (if it exists) and we shall see that this is independent of the surfaces.

Much of the time it is convenient to abbreviate $c(\overbrace{x\dots x}^n,y)$ as $c(n)$ so $c(x,y)=c(1)$.

If $c(n)$ is defined and rationally null-homologous then $\b^n(x,y)$ is defined to be the set of rational numbers ${\ensuremath{\ell k}}(c(n),c^+(n))$, ranging over all possible ways of defining such a $c(n)$. If no such $c(n)$ exists then $\b^n(x,y)$ is undefined.

\[Example3\] Consider the manifold $M$, shown in Figure \[example3\], obtained from $0$-framed surgery on a two component link $\{ L_x,L_y\}$. Use a genus one Seifert surface for $L_y$ obtained from the obvious twice-punctured disk and a tube that goes up to avoid $L_x$. Let $V_y$ be this surface capped-off in $M$. Similarly use the fairly obvious Seifert surface for $L_x$ in the complement of $L_y$. Then $c^+(x,y)$ is shown. Since it has self-linking zero with respect to $V_x$, $\b^1(x,y)=\b^1(y,x)=0$. Furthermore $\b^2(x,y)=-1$ (note the link $\{c