ScalingStacks

Proof. We first show independence of $\phi$, given a fixed choice of $\phi_{\mathrm{in}}$ and $\phi_{\mathrm{out}}$. Suppose that $\phi^{\prime}\colon W\to{\mathbb{R}}^{3}\times{[0,1]}$ is another diffeomorphism restricting to $\phi_{\mathrm{in}}$ and $\phi_{\mathrm{out}}$ on $W_{\mathrm{in}}$ and $W_{\mathrm{out}}$ respectively.

Lemma 4.7, proved below, implies that the link cobordisms $\phi(\Sigma)$ and $\phi^{\prime}(\Sigma)$ are isotopic rel boundary in ${\mathbb{R}}^{3}\times{[0,1]}$ and we have $\mathrm{KhR}_{N}(\phi(\Sigma))=\mathrm{KhR}_{N}(\phi^{\prime}(\Sigma))$ by Theorem 2.4.

Next we show independence of $\phi_{\mathrm{in}}$, given a fixed choice of $\phi_{\mathrm{out}}$. Let $\phi^{\prime}_{\mathrm{in}}\colon W_{\mathrm{in}}\to{\mathbb{R}}^{3}$ be another diffeomorphism such that $\phi^{\prime}_{\mathrm{in}}(\Sigma_{\mathrm{in}})$ is generic and blackboard-framed, and $\phi^{\prime}$ another diffeomorphism $W\to{\mathbb{R}}^{3}\times{[0,1]}$ restricting to $\phi^{\prime}_{\mathrm{in}}$ on $W_{\mathrm{in}}$ but still to $\phi_{\mathrm{out}}$ on $W_{\mathrm{out}}$. Then, by Lemma 4.1 (1), we can find a family $\phi_{\mathrm{in},t}$ connecting $\phi_{\mathrm{in}}$ to $\phi^{\prime}_{\mathrm{in}}$. By definition of parallel transport, we have:

$$\eta(\phi^{\prime}_{\mathrm{in}})=\mathrm{KhR}_{N}(\phi_{\mathrm{in},t}(\Sigma_{\mathrm{in}}))(\eta(\phi_{\mathrm{in}}))$$

Now we obtain a new diffeomorphism $\phi^{\prime}\circ\phi_{\mathrm{in},t}\colon W\to{\mathbb{R}}^{3}\times{[0,1]}$ and by the previous independence result and Theorem 2.4, we have:

$$\mathrm{KhR}_{N}(\phi^{\prime}(\Sigma))(\eta(\phi^{\prime}_{\mathrm{in}}))=\mathrm{KhR}_{N}(\phi^{\prime}(\Sigma)\circ\phi_{\mathrm{in},t}(\Sigma_{\mathrm{in}}))(\eta(\phi_{\mathrm{in}}))=\mathrm{KhR}_{N}(\phi(\Sigma))(\eta(\phi_{\mathrm{in}}))$$

Thus, the definition was independent of the choice of $\phi_{\mathrm{in}}$. An analogous argument also establishes independence of the choice of $\phi_{\mathrm{out}}$. ∎