Subsection 3.2. (0NC0) · ScalingStacks

3.2. The game plan

Fix an almost closure $T$ of a braid word $\beta$ for $[\beta]\in\mathrm{Br}_{n+1}$ . We call the right-hand closure $L$ and the left-hand closure $L^{\prime}$ . We consider the following movies of intermediate diagrams and their associated chain maps between Khovanov–Rozansky complexes.

(3.1)

In the first row, the $\pm$ signs indicate the two versions of this movie, in which the horizontal strand passes in front of ( $+$ ) or behind ( $-$ ) $T$ . We denote the composition along the top by $\mathrm{sw}_{+}$ and the composition along the bottom by $\mathrm{sw}_{-}$ . In either case we first see a Reidemeister I move (denoted by $R1_{\pm}$ ), then a composite of Reidemeister II moves (denoted by $R2_{\pm}$ ), a number of Reidemeister III moves (each denoted by $R3_{\pm}$ ), a composite of inverse Reidemeister II moves ( $R2^{-1}_{\pm}$ ), and finally an inverse Reidemeister I move ( $R1^{-1}_{\pm}$ ). Our goal is to show that, after making careful use of the freedom, described later, to choose up-to-homotopy representatives of the chain maps for Reidemeister III moves, we have the following:

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Theorem 3.3. For every almost braid closure diagram $T$ , the front sweep $\mathrm{sw}_{+}$ and the back sweep $\mathrm{sw}_{-}$ chain maps constructed above are identical (not just merely homotopic).

Together with Proposition 3.2, this will imply Theorem 1.1.

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Corollary 3.4. For every almost braid closure diagram $T$ , we have $\mathrm{sw}_{T}=\mathbf{1}_{T}$ .

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Proof. We have $\mathrm{sw}_{T}=(\mathrm{sw}_{-})^{-1}\circ\mathrm{sw}_{+}=(\mathrm{sw}_{-})^{-1}\circ\mathrm{sw}_{-}=\mathbf{1}_{T}$ . ∎

The proof of Theorem 3.3 will occupy the rest of this section.

We distinguish two types of crossings in the intermediate diagrams $L_{\pm}^{i}$ . The crossings of the moving, horizontal, strand with everything else will be called external. The remaining crossings were already present in $T$ and will be called internal.

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Definition 3.5. The homological grading on $\left\llbracket L^{i}_{\pm}\right\rrbracket$ splits into the sum of the internal and external homological gradings, contributed by resolutions of internal and external crossings respectively. The internal and external homological degrees of a web $W$ appearing in $\left\llbracket L^{i}_{\pm}\right\rrbracket$ will be denoted by $\mathrm{gr}_{\mathrm{int}}(W)$ and $\mathrm{gr}_{\mathrm{ext}}(W)$ respectively.

The braid word $\beta$ determines an ordering of the crossings in $T$ , $L$ , and $L^{\prime}$ , namely from top to bottom. This ordering also induces an ordering of the internal crossings in all other diagrams in (3.1). The diagrams $L^{0}_{\pm}$ and $L^{x}_{\pm}$ have one additional external crossing. The diagrams $L^{i}_{\pm}$ for $1\leq i\leq x-1$ all have $2n+1$ external crossings, which are ordered from right to left. We will classify webs $W$ in each of these complexes according to the resolutions that appear at the crossings. For the following, let $M$ denote the number of crossings in $T$ , $L$ , and $L^{\prime}$ .

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Definition 3.6. The type of a web $W$ in any of the complexes in (3.1) is the element $\tau(W)\in\{p,t\}^{M}$ that records in the $j$ -th coordinate whether the $j$ -th internal crossing in the respective link diagram is resolved in a parallel way (p), or using the thick edge (t).

The offset of a web $W$ in any of the complexes $\left\llbracket L^{i}_{\pm}\right\rrbracket$ is the element $o(W)\in\{p,t\}$ that records the resolution of the leftmost external crossing.

The state of a web $W$ in any of the complexes $\left\llbracket L^{i}_{\pm}\right\rrbracket$ for $1\leq i\leq x-1$ is the element $s(W)\in\{p,t\}^{2n}$ , which records the resolutions of the $2n$ rightmost external crossings (that is, all except the leftmost external crossing). Such a web $W$ is said to be palindromic if $s(W)$ is a palindrome.

Remark. The webs $W$ in the complexes $\left\llbracket L\right\rrbracket$ and $\left\llbracket L^{\prime}\right\rrbracket$ are indexed by their types $\tau(W)$ . The webs $W$ in $\left\llbracket L^{0}_{\pm}\right\rrbracket$ , and $\left\llbracket L^{x}_{\pm}\right\rrbracket$ are indexed by the pairs $(\tau(W),o(W))$ . The webs $W$ in the complexes $\left\llbracket L^{i}_{\pm}\right\rrbracket$ for $1\leq i\leq x-1$ are indexed by the triples $(\tau(W),o(W),s(W))$ .

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Definition 3.7. If $i\in\{0,1,\cdots,x\}$ , $\epsilon\in\{+,-\}$ , $s\in\{p,t\}^{2n}$ , $o\in\{p,t\}$ and $\tau\in\{p,t\}^{M}$ , we will use $W_{\epsilon}^{i}(\tau,o,s)$ or $W_{\epsilon}^{i}(\tau,o)$ to denote the web in $\left\llbracket L_{\epsilon}^{i}\right\rrbracket$ with indexing data $(\tau,o,s)$ or $(\tau,o)$ , as appropriate. Analogously, we write $W(\tau)$ and $W^{\prime}(\tau)$ for $\tau$ -indexed webs in $\left\llbracket L\right\rrbracket$ and $\left\llbracket L^{\prime}\right\rrbracket$ respectively. If the indexing data is fixed, we will sometimes omit it from the notation (e.g. $W^{i}_{\pm}=W^{i}_{\pm}(\tau,o,s)$ and $W=W(\tau)$ ) and say that the webs $W^{i}_{+}$ and $W^{i}_{-}$ correspond to each other.

If $f$ is a chain map and $V$ and $W$ are webs in the source and target complexes, then we write $f(V,W)$ for the component of $f$ from $V$ to $W$ .

Remark. Suppose $s\in\{p,t\}^{2n}$ , $o\in\{p,t\}$ and $\tau\in\{p,t\}^{M}$ . For $1\leq i\leq x-1$ we have $W^{i}_{+}(\tau,o,s)=W^{i}_{-}(\tau,o,s)$ as webs, and for $i\in\{0,x\}$ we have $W^{i}_{+}(\tau,o)=W^{i}_{-}(\tau,o)$ as webs. Moreover, $\mathrm{gr}_{\mathrm{ext}}(W^{i}_{+}(\tau,p,s))=-\mathrm{gr}_{\mathrm{ext}}(W^{i}_{-}(\tau,p,s))$ .

Scott Morrison, Kevin Walker, Paul Wedrich

Original source: arXiv:1907.12194v5