Knots and entanglement

1. Big piece of work with in-depth discussion of many aspects

2. Original idea and approach

1. The paper is too long and technical

2. I do not think it fits the readership of SciPost Physics

Our understanding of topological entanglement entropy in two spatial dimension [6] is the culmination of decades of research on the Chern-Simons theory and its relation with conformal field theory, knots and the geometry of three-manifolds. Many deep mathematical aspects entering these theories have eventually become understandable by a large theoretical physics audience. Therefore, there are many papers on this subject in physics journal, also discussing experimental and numerical verifications. The situation in one dimension higher is very different.

This paper originates from an abstract reformulation of known properties of 2d entanglement by some some of the authors and extends this approach to 3d. It is rather clear that this brave step is very technical and should be first proposed to a mathematical physics audience, writing a paper in the language appropriate for it and submitting it to e.g. Commun. Math. Phys., not to SciPost Physics.

The paper is not only theorems and lemmas. There is an example in Section 4, the Kitaev quantum double model, where this approach is put in practice.

I do not think this paper is appropriate for SciPost Physics.

If I were the authors, I would write another paper, that contains Section 4 only, the idea being that of introducing concepts from the example. I would refer to the longer math-oriented paper for proofs. This other paper would be definitely short, could be appropriate for SciPost and would let many colleague appreciate the novel approach to entanglement.

1- This manuscript extends the authors' previous work on entanglement bootstrap from two to three spatial dimensions. It is an independent and distinguished framework, comparing to parallel approaches.

2- This work provides a meaningful connection between quantum information, in particular, quantum entanglement, and liquid topological ordered phases.

3- Most of the physical results obtained are reasonable.

1- Unclear, inaccurate writing, especially in Section 2. I list a few examples below.
(1) A few lines above Example 2.11, $I$ was for an interval, but in and after Example 2.11, $I$ is for sector label, without any clarification. Moreover, $I=\{a,b,c\}$ ought to be an ordered triple instead of merely a set. This abuse of notation also happens multiple times elsewhere, and can be more confusing, for example in Proposition 2.18.
(2) Proposition 2.22 "Let $\Omega$ be a region embedded in $S$", some conditions seem missing for such embedding, since based on the proof, $\Omega$ can not share boundary regions with $S$.
(3) $\eta^\vee$ in Figure 8 is confusing. According to Point 3 on Page 31, $^\vee$ is applied to the label of the entire Hopf-link excitation.
(4) Point 4 on Page 32, below equation (3.10) "Deform BC to AB by a path" seems to flip the interior to the exterior of a solid torus. Can the author be more explicit about the path, which is quite counterintuitive?

2- Not written in a self-contained way. Poor organization. The authors frequently omit proofs similar to those in earlier works, or postpone proofs and details to the appendix. This style is acceptable, however, given the current length of the manuscript, not necessary. In most cases it does not make the storyline more smooth and adds pain to reading.

3- This part is related to point 1- and 2-, but let me put it as a standalone point for emphasis: The notion of immersion does not receive serious treatment (or maybe it does, but not clear exposition):
(1) Definition 2.1. What is the precise meaning of "lift" and "layered regions"? Why is it always possible to lift?
(2) In the followings, since the authors kept omitting proofs while referring to their previous work, and keeping only one datum $\Omega$ while dropping $\Omega^+$ and $\mathfrak p$, I cannot find the very detail about, and thus get convinced that "We generalize the information convex set to a new class of regions called immersed regions, promoting various theorems to this new context" as claimed in the abstract.
(3) In most later applications after Section 2, the authors simply write "regions". It is unclear where the immersion really plays a role.

The article suffers from an awkward balance between mathematical rigor and physical ideas. The mathematical statements should be, at least, precise enough for experts to follow and repeat. The manuscript failed to reach such a point. However, the authors seem to have good enough intuitions and pictures underneath, by which they obtain quite reasonable physical results. A recommendation for publication should be promising if the authors could make the exposition more clear and precise.

1- Duplicated references [32] and [50].