Subsection 3.1. (0NBZ) · ScalingStacks

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 $\mathrm{sw}_{T}\simeq\textrm{id}_{T}$ . ∎

Scott Morrison, Kevin Walker, Paul Wedrich

Original source: arXiv:1907.12194v5