ScalingStacks

Proof of Theorem 3.3. We need to show that the two chain maps $\mathrm{sw}_{+}$ and $\mathrm{sw}_{-}$ from (3.1) are equal. For this, let $W$ and $W^{\prime}$ be webs in $\left\llbracket L\right\rrbracket$ and $\left\llbracket L^{\prime}\right\rrbracket$ respectively. We shall compare the components of $\mathrm{sw}_{+}$ and $\mathrm{sw}_{-}$ between $W$ and $W^{\prime}$.

By Proposition 3.13, the $R3_{\pm}$ maps do not decrease the external homological degree, but by Lemmas 3.8 and 3.10, the $R1^{\pm 1}_{\pm}$ and $R2^{\pm 1}_{\pm}$ maps preserve the external homological degree. Since $\mathrm{gr}_{\mathrm{ext}}(W)=\mathrm{gr}_{\mathrm{ext}}(W^{\prime})=0$, the increasing components of $R3_{\pm}$ do not contribute to $\mathrm{sw}_{+}$ or $\mathrm{sw}_{-}$. Now suppose that $W^{1}_{\pm}$ are corresponding webs in $\left\llbracket L^{1}_{\pm}\right\rrbracket$ and $W^{x-1}_{\pm}$ are corresponding webs in $\left\llbracket L^{x-1}_{\pm}\right\rrbracket$ with $\mathrm{gr}_{\mathrm{ext}}(W^{1}_{\pm})=\mathrm{gr}_{\mathrm{ext}}(W^{x-1}_{\pm})=0$. Then, by Corollary 3.14,

$$(R3_{+}\circ\cdots\circ R3_{+})(W^{1}_{+},W^{x-1}_{+})=(R3_{-}\circ\cdots\circ R3_{-})(W^{1}_{-},W^{x-1}_{-}).$$

Let us also record that if $R3_{\pm}\circ\cdots\circ R3_{\pm}$ has a non-zero component between two webs $W^{1}$ and $W^{x-1}$, then first $n$ digits of $t(W^{1})$ agree with the first $n$ digits of $t(W^{x-1})$. (Recall that the first $n$ digits describe the rightmost $n$ crossings, which are spatially separated from the region in which Reidemeister III moves occur.)

Next we consider the pair of corresponding webs $W^{0}_{\pm}=W^{0}_{\pm}(s(W),p)$ in $\left\llbracket L^{0}_{\pm}\right\rrbracket$, which appear in the image of $W$ under $R1_{\pm}$, and the pair of corresponding webs $W^{x}_{\pm}=W^{x}_{\pm}(s(W^{\prime}),p)$ in $\left\llbracket L^{x}_{\pm}\right\rrbracket$, which have $W^{\prime}$ as image under $R1^{-1}_{\pm}$. The components of $R2^{-1}_{\pm}\circ R3_{\pm}\circ\cdots\circ R3_{\pm}\circ R2_{\pm}$ between these webs are sums over components through many possible intermediate webs $W^{1}_{\pm}$ and $W^{x-1}_{\pm}$. By the previous argument, the Reidemeister III portions of the $+$- and the $-$-version of the map agree. By Lemma 3.11, the Reidemeister II portions could at most cause a sign-discrepancy. However, since the first $n$ digits of $t(W^{1}_{\pm}),t(W^{2}_{\pm}),\ldots,t(W^{x-1}_{\pm})$ all agree, and since Reidemeister II chain maps are zero on non-palindromic webs by Lemma 3.10, there is no sign-discrepancy. Thus, we record:

$$(R2^{-1}_{+}\circ R3_{+}\circ\cdots\circ R3_{+}\circ R2_{+})(W^{0}_{+},W^{x}_{+})=(R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-})(W^{0}_{-},W^{x}_{-})$$

Finally, we use Lemma 3.9 to compute:

	$\displaystyle\mathrm{sw}_{-}(W,W^{\prime})$	$\displaystyle=(R1^{-1}_{-}\circ R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}\circ R1_{-})(W,W^{\prime})$
		$\displaystyle=(R1^{-1}_{-}\circ R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}\circ p\circ R1_{+})(W,W^{\prime})$
		$\displaystyle=(R1^{-1}_{-}\circ p^{\prime}\circ R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}\circ R1_{+})(W,W^{\prime})$
		$\displaystyle=(R1^{-1}_{+}\circ R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}\circ R1_{+})(W,W^{\prime})$
		$\displaystyle=(R1^{-1}_{+}\circ R2^{-1}_{+}\circ R3_{+}\circ\cdots\circ R3_{+}\circ R2_{+}\circ R1_{+})(W,W^{\prime})$
		$\displaystyle=\mathrm{sw}_{+}(W,W^{\prime})\qed$