report_X_on_v1:

This is a really interesting paper that raises many cool questions and conjectures, but I think it's not at the current level of JEMS.  Overall, I'd say this should appear in something like AGT or Exp. Math, with the latter being a particularly natural venue.  

To recap, the authors introduce a polynomial invariant of knots that can be computed quickly.  They conjecture it is the 2-loop invariant of Ohtsuki and gives a lot of conjectured properties.  The paper is written in the first author's inimitable style, and feels a bit abortive in places.  For example:

1) Despite their repeated assertions that their invariant is fast to compute, there is no actual analysis of its running time.  In Comment 5, they estimate the number of ring operations in Q(T) needed, but such operations are not constant time.  Indeed, a naive implementation of that ring will have problems with denominator blow up.  I think p-adic methods mean you can actually compute their theta in polynomial time --- this should all be in the computational literature, the ideas go back to Dixon --- but it would be nice to work this out.

2) In Section 5.2.1, they note that assuming Conjecture 18 then the 48-crossing GST knot must have genus at least 10.  Szabó's HFK calculator confirms in less than half a second that the genus is exactly 10.

report_Y_on_v1:

The invariant the authors constructed has a symmetry of the root system of sl3.  So the invariant must be related to the sl3 version of the Alexander polynomial constructed by Harper arXiv:2008.06983.  If some relation to Harper's invariant is mentioned, this paper would be good to be considered.

report_Z_on_v1:

The paper under consideration is highly unusual. The authors construct
a knot invariant, with potential motivation from at least two
directions (the loop expansion from finite type invariants, or from
perturbed quantization), but prove none of these connections; the
definition they end up with is formulas that they fully acknowledge
are "uninspiring" (p. 6) and with coefficients determined by the
undetermined coefficients method (Comment 34). Although they assert an
independence proof (which the reviewer has not yet checked), they are
not able to prove even the most basic and obvious facts on examination
of the invariant, like the hexagonal invariance. It is highly likely
this definition of the invariant will not last very long as the core
definition.

But they nevertheless argue convincingly that their invariant is fast,
strong, topologically meaningful, and fun, all four of which are key
properties. There are really intriguing patterns in their data that
are calling out stringly for study and explanation. I am personally
particularly intrigued by the patterns reminiscent of Chladni figures
appearing in the coefficents, most evident in Figure 1.4, although
this particular feature is probably well down the road on the study of
the invariant.

To me this is one of the most exciting developments in knot theory in
many years, in that it is pointing to a new world to explore with a
lot to explain. There are a huge variety of conjectures and questions
resulting from this invariant.

I highly recommend giving this paper a full review.