Subsection 3.5. (0NC3) · ScalingStacks

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.

Scott Morrison, Kevin Walker, Paul Wedrich

Original source: arXiv:1907.12194v5