\{0\}\in S^{1}\times\{0\}$ maps to $y\in S^{2}$, $x\times
\{1\}\in S^{1}\times\{1\}$ maps to $-y$. Let us now define $\beta_{0}$ as a map that maps the border of a disc to $\pm\dot{\gamma}$, with the sign corresponding to that of $\overline{\Phi_{H}}(S^{1}\times \{0\})$, and $\beta_{1}$ to be $-\beta_{0}$. By abuse of notation, let us now define our total map $\beta$ as the difference between these two maps. Our correction term will then be defined by the equation $$\tau_{\beta}(\gamma):=\frac{1}{2\pi}\int_{D^{2}}\beta^{*}\omega$$ The motivation for this is that just like with $\gamma$, invariance of $\tau_{\beta}$ means that $\tau_{\beta , {0}}(\gamma)=\tau_{\beta
,{1}}(\gamma)$ for all $\gamma$. But: $$0=\int\limits_{D^{2}\times I} d\beta^{*}\omega=
\int\limits_{\partial(D^{2}\times I)}\beta^{*}\omega= \tau_{\beta
,{1}}-\tau_{\beta ,{0}}+2\int\limits_{S^{1}\times I}
\beta^{*}\omega \label{motv}$$ So $\tau_{\beta}$ is “at a distance” of $2\int\limits_{S^{1}\times I} \beta^{*}\omega$ from being invariant.

We have found that $\partial\tau_{\beta}=\partial\eta$, proving the following:

$\eta-\tau_{\beta}$ is an invariant of ordered pairs of a knot and an integer specifying the homotopy class for the map $\beta$.

Indeed, a simple Stokes’ theorem argument shows that for two homotopic $\beta$’s give us the same $\tau_{\beta}$ (we can push the difference to the boundary, where the two $\beta$’s will coincide). Moreover, our new invariant assumes integer values, because $\beta$ wraps the disc around $S^{2}$ a whole number of times (it 