Section 3. (0NBY) · ScalingStacks

3. The sweep-around move

The purpose of this section is to prove Theorem 1.1.

3.1. Reduction to almost braid closures

Given a braid word $\beta$ for a braid $[\beta]\in\mathrm{Br}_{n+1}$ , we can get a 1-1-tangle diagram by taking the braid closure of the $n$ rightmost strands. We say that such 1-1-tangle diagrams are in almost braid closure form. From a 1-1-tangle diagram $T$ , one can obtain link diagrams $L$ and $L^{\prime}$ by taking either the left- or right-handed closure of the single open strand. These diagrams are illustrated at the top left and top right of (3.1) respectively.

We note the following straightforward extension of the Alexander theorem.

0NA1

Lemma 3.1. Every 1-1-tangle can be isotoped into almost braid closure form.

0NA2

Proposition 3.2. If the sweep-around map is homotopic to the identity for 1-1-tangles in almost braid closure form, then the same is true for all 1-1-tangle diagrams.

0NA3

Proof. Consider an isotopy that brings the tangle diagram $T$ into almost braid closure form $T^{\prime}$ and denote its image under the Khovanov invariant as $\phi$ . Furthermore, let the maps associated to the sweep-around for $T$ and $T^{\prime}$ be denoted by $\mathrm{sw}_{T}$ and $\mathrm{sw}_{T^{\prime}}$ respectively. Now, note that $\mathrm{sw}_{T}\simeq\phi^{-1}\circ\mathrm{sw}_{T^{\prime}}\circ\phi$ because the underlying link cobordisms are isotopic in ${\mathbb{R}}^{3}\times{[0,1]}$ . By assumption $\mathrm{sw}_{T^{\prime}}\simeq\textrm{id}_{T^{\prime}}$ and thus also $\mathrm{sw}_{T}\simeq\textrm{id}_{T}$ . ∎

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:

0NA4

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.

0NA5

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

0NA6

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.

0NA7

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 $L^{\prime}$ .

0NA8

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 $(\tau(W),o(W),s(W))$ .

0NA9

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, $\mathrm{gr}_{\mathrm{ext}}(W^{i}_{+}(\tau,p,s))=-\mathrm{gr}_{\mathrm{ext}}(W^{i}_{-}(\tau,p,s))$ .

3.3. Reidemeister I moves

The Reidemeister I chain maps are the following.

Here $\mathrm{cap}$ and $\mathrm{cup}$ simply denote the cap and cup foams, while $\mathrm{dcap}$ and $\mathrm{dcup}$ denote decorated cap and cup foams. The decoration is by the polynomial $\sum_{a+b=N-1}{X^{a}Y^{b}}$ where $X$ denotes the dot on the strand and $Y$ the dot on the circle; see (2.2). We have only assigned notation $R1_{\pm}$ and $R1^{-1}_{\pm}$ to those Reidemeister I chain maps that are relevant for the sweep-around move.

0NAA

Lemma 3.8. The Reidemeister I chain maps $R1_{\pm}\colon\left\llbracket L\right\rrbracket\to\left\llbracket L^{0}_{\pm}\right\rrbracket$ and $R1^{-1}_{\pm}\colon\left\llbracket L^{x}_{\pm}\right\rrbracket\to\left\llbracket L^{\prime}\right\rrbracket$ preserve the internal and external homological degrees individually. Moreover, their only non-zero components are in external homological grading zero.

0NAB

Proof. This is immediate from the explicit description of these chain maps. ∎

0NAC

Lemma 3.9. In external homological grading zero, we have $R1_{-}=p\circ R1_{+}$ and $R1^{-1}_{+}=R1^{-1}_{-}\circ p^{\prime}$ where $p$ and $p^{\prime}$ are chain maps of decorated identity foams such that

R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}\circ p=p^{\prime}\circ R2^{-1}_{-}\circ R3_{-}\circ\cdots\circ R3_{-}\circ R2_{-}

0NAD

Proof. The chain maps $p$ and $p^{\prime}$ each consist of identity foams decorated by the polynomial $\sum_{a+b=N-1}{X^{a}Y^{b}}$ . In $p$ , the dots $X$ and $Y$ are placed next to the Reidemeister I crossing, as shown in the first picture on the right. These dots are spatially separated from the region in which the $R2$ and $R3$ moves are taking place, so we can slide them spatially lower in the diagram, and timewise past all the $R2$ and $R3$ moves. 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3.4. Reidemeister II moves

We will use Elias–Khovanov’s Soergel calculus [EK10a] to describe the chain maps associated to Reidemeister II and III moves. The Soergel calculus of type $A_{n-1}$ is a graphical incarnation of the 2-category of Soergel bimodules, which categorifies the Hecke algebra for $S_{n}$ . For any $N\geq 2$ , it admits a 2-functor to the monoidal subcategory of $\boldsymbol{\mathrm{Foam}}_{N}$ of webs and foams with $2n$ boundary components with suitable orientations, see e.g. [MV10]. Instead of describing these $2$ -functors formally, we will just use the Soergel calculus as shorthand notation for foams using the following dictionary:

•

In the $A_{1}$ calculus, we have only a blue object, which we will interpret as the two strand web

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•

In the $A_{2}$ calculus, we have red and blue objects, interpreted as three strand webs

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•

Start dots and end dots (in any color) correspond to zip and unzip foams.
•

The trivalent vertices and correspond to digon creation and annihilation foams respectively. 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•

The 6-valent vertex corresponds to the foam shown in Figure 1.

The Reidemeister II chain maps are the following.

(3.2)

In both cases we have chosen to order the crossings from the top to the bottom. Now we can record two observations concerning the composite (inverse) Reidemeister II chain maps $R2_{\pm}$ and $R2^{-1}_{\pm}$ .

0NAE

Lemma 3.10. The chain maps $R2_{\pm}\colon\left\llbracket L^{0}_{\pm}\right\rrbracket\to\left\llbracket L^{1}_{\pm}\right\rrbracket$ and $R2^{-1}_{\pm}\colon\left\llbracket L^{x-1}_{\pm}\right\rrbracket\to\left\llbracket L^{x}_{\pm}\right\rrbracket$ preserve the internal and external homological gradings individually and their only non-zero components involve palindromic resolutions.

0NAF

Lemma 3.11. Let $W^{0}_{\pm}=W^{0}_{\pm}(\tau,o)$ and $W^{x}_{\pm}=W^{x}_{\pm}(\tau,o)$ be pairs of corresponding webs in $\left\llbracket L^{0}_{\pm}\right\rrbracket$ and $\left\llbracket L^{x}_{\pm}\right\rrbracket$ respectively. Further, let $s\in\{p,t\}^{2n}$ be a palindrome in which $t$ appears $2k$ times, and consider $W^{1}_{\pm}=W^{1}_{\pm}(\tau,o,s)$ and $W^{x-1}_{\pm}=W^{x-1}_{\pm}(\tau,o,s)$ in $\left\llbracket L^{1}_{\pm}\right\rrbracket$ and $\left\llbracket L^{x-1}_{\pm}\right\rrbracket$ respectively. Then

	$\displaystyle R2_{-}(W^{0}_{-},W^{1}_{-})$	$\displaystyle=(-1)^{k}R2_{+}(W^{0}_{+},W^{1}_{+})$
	$\displaystyle R2^{-1}_{+}(W^{x-1}_{+},W^{x}_{+})$	$\displaystyle=(-1)^{k}R2^{-1}_{-}(W^{x-1}_{-},W^{x}_{-}).$

0NAG

Proof. In a single Reidemeister II move, the identity resolution is always sent to the identity resolution via the identity. The maps involving the resolution with two thick edges are negatives of each other, when comparing the two types of Reidemeister II moves with fixed order of crossings as in (3.2). ∎

3.5. Reidemeister III moves

In (3.1) we encounter four types of Reidemeister III moves. Namely, the moving strand can pass in front of or behind a positive or a negative crossing. In the following we show the front and back versions alongside each other. In every case, the moving strand is the one connecting the bottom left and top right boundary points.

In each variant of Reidemeister III, we order the crossings in each tangle from top to bottom. The parts of the complexes with internal homological degree zero—where the internal crossing is resolved in the parallel fashion—are highlighted in blue. The parts with internal homological degree $\pm 1$ are highlighted in yellow.

There is a 2-dimensional space of chain maps between the two sides of each Reidemeister III move [EK10b]. There is a 1-dimensional affine subspace of these chain maps which, given the previous choices for Reidemeister I and II maps, provides a functorial link invariant, by Theorem 2.4. (Note that their proof does not rely on any particular choice of chain maps from this subspace; any will do!) This subspace is characterised by the condition that the component of the chain map between parallel resolutions is the identity (this condition corresponds to the appearance of a blue highlighted $1$ in each chain map below). In the diagrams below, we parametrise this subspace by a variable $y$ ; shortly we shall specialize to $y=0$ .

All choices of chain map in this affine subspace are homotopic, so for many purposes this structure can be ignored. For the present proof, however, it is quite important that we make the most convenient choice of up-to-homotopy representative.

When the moving strand passes a positive crossing we have:

(3.3)

Next, we consider the two ways in which the moving strand may pass a negative crossing:

(3.4)

For the remainder of this paper we specialise to the choice $y=0$ . (Note in particular that the statements immediately below are not true for other choices!)

0NAH

Lemma 3.12. The chain maps $R3_{\pm}\colon\left\llbracket L^{i}_{\pm}\right\rrbracket\to\left\llbracket L^{i+1}_{\pm}\right\rrbracket$ in (3.1) do not decrease the external homological grading.

0NAI

Proof. Since chain maps are of homological degree zero, the statement is equivalent to saying that the Reidemeister III chain maps in (3.1) never increase the internal homological grading. This can be verified by inspecting (3.3) and (3.4). For the reader’s convenience we have highlighted the components of negative internal homological degree in green. All other non-zero components are highlighted blue or yellow and have internal homological degree zero because they map between the yellow and blue layers of the relevant complexes. Thus we only need to worry about components of the chain map which are not highlighted in the diagrams above. With $y=0$ , these components all vanish. ∎

In other words, the Reidemeister III maps are filtered with respect to the filtration determined by the internal homological degree, which we shall call the internal filtration.

0NAJ

Proposition 3.13. The filtration-preserving components of the chain maps

R3_{+}\colon\left\llbracket L^{i}_{+}\right\rrbracket\to\left\llbracket L^{i+1}_{+}\right\rrbracket\quad\text{and}\quad R3_{-}\colon\left\llbracket L^{i}_{-}\right\rrbracket\to\left\llbracket L^{i+1}_{-}\right\rrbracket

agree if $1\leq i<x-1$ . More precisely, we have

R3_{+}(W^{i}_{+},W^{i+1}_{+})=R3_{-}(W^{i}_{-},W^{i+1}_{-})

for pairs of corresponding webs $W^{i}_{\pm}$ in $\left\llbracket L^{i}_{\pm}\right\rrbracket$ and $W^{i+1}_{\pm}$ in $\left\llbracket L^{i+1}_{\pm}\right\rrbracket$ with $\mathrm{gr}_{\mathrm{ext}}(W^{i}_{\pm})=\mathrm{gr}_{\mathrm{ext}}(W^{i+1}_{\pm})$ .

0NAK

Proof. By inspecting (3.3) and (3.4) — for each of the 1+9+9+1 components of the $R3_{+}$ chain map, check that the corresponding component of the $R3_{-}$ chain map is the same (recalling $y=0$ ). ∎

0NAL

Corollary 3.14. The filtration-preserving component of the chain maps

R3_{+}\circ\cdots\circ R3_{+}\colon\left\llbracket L^{1}_{+}\right\rrbracket\to\left\llbracket L^{x-1}_{+}\right\rrbracket\quad\text{and}\quad R3_{-}\circ\cdots\circ R3_{-}\colon\left\llbracket L^{1}_{-}\right\rrbracket\to\left\llbracket L^{x-1}_{-}\right\rrbracket

agree. More precisely, we have

R3_{+}(W^{1}_{+},W^{x-1}_{+})=R3_{-}(W^{1}_{-},W^{x-1}_{-})

for pairs of corresponding webs $W^{1}_{\pm}$ in $\left\llbracket L^{1}_{\pm}\right\rrbracket$ and $W^{x-1}_{\pm}$ in $\left\llbracket L^{x-1}_{\pm}\right\rrbracket$ with $\mathrm{gr}_{\mathrm{ext}}(W^{1}_{\pm})=\mathrm{gr}_{\mathrm{ext}}(W^{x-1}_{\pm})$ .

Remark. The Reidemeister III chain maps shown in (3.3) and (3.4), their inverses, and four additional variations were studied by Elias–Krasner [EK10b]. Note, however, the following differences in conventions. Their positive crossings are our negative crossings and the crossings in their braids are ordered from bottom to top, while we order them from top to bottom. Finally, they read Soergel diagrams from left to right, while we read them from right to left.

3.6. Proof of the sweep-around property

0NAM

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:

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

Scott Morrison, Kevin Walker, Paul Wedrich

Original source: arXiv:1907.12194v5