Peer review of: Creation of acoustic vortex knots (2020)

Reviewers' Comments: 
 
Reviewer #1: 
Remarks to the Author: 

The paper presents theoretical and experimental realizations of acoustic knotted vortices. I think the 
experimental measurements of these fragile and intricate fields are enough to warrant publication. I 
am sure that people working on knots (being them optic, plasma, water flow...) will appreciate this 
achievement. The paper is nicely written, to the point and properly motivated. That is why I would 
recommend the paper for acceptance. 
 
However, I would like more information on how the knots where detected once that the acoustic field 
was scanned. In the paper, the authors talk about "dark points", I imagine that these are points at 
which theoretically the amplitude is 0. But, I imagine that experimentally there are some details about 
how to get this. Do the authors look for minima of amplitude? In any case, I am not convinced by this 
method. Most of the papers on knotted vortices that I have seen look in the phase and check for 
vortices using algorithms based on the phase. No plots of the phase appear in the main paper. 
 
Trapping of particles in acoustic vortices or shaped fields is described but no reference is provided, I 
would recommend these: 
- Marzo, A., Seah, S. A., Drinkwater, B. W., Sahoo, D. R., Long, B., & Subramanian, S. (2015). 
Holographic acoustic elements for manipulation of levitated objects. Nature communications, 6, 8661. 
- Baresch, D., Thomas, J. L., & Marchiano, R. (2016). Observation of a single-beam gradient force 
acoustical trap for elastic particles: acoustical tweezers. Physical review letters, 116(2), 024301. 
 
 
Reviewer #2: 
Remarks to the Author: 
This paper reports on theoretically and experimentally determined acoustic vortex knots (a Hopf link 
and Trefoil knot) using a metamaterial aperture. Design of the metamaterial was guided by theoretical 
formulations from topology for the particular types of vortex link and knot examined. The 
metamaterial aperture consists of an arrangement of 24 x 24 units (pixels), that was fabricated using 
SLA 3D printing. Each unit cell was numerically optimized and consisted of a coating unit cell and 
perforated panels that provided modulation of both the magnitude and phase. Acoustic testing was 
performed, and the 2D scans of sound pressure magnitude were taken at multiple distances from the 
metamaterial aperture, with results showing a clear demonstration of an acoustic link and acoustic 
knot. 
 
This paper is well-written, and clearly and concisely describes the work presented. While vortex waves 
and vortex knots have been examined extensively in other fields of science (such as optics), it has to 
this reviewers knowledge never been experimentally demonstrated in acoustics before. As a result, 
this work represents a significant and novel contribution to the scientific literature on this topic. 
Furthermore, the authors were thorough in their analysis, utilizing both theoretical and numerical 
simulation and optimization techniques, and even performing component level experiments examining 
the viscous effects of air on their metamaterial design (as described in the supplementary section). 
 
Although overall very good, several minor issues were identified with the paper which should be 
addressed: 
 
1) Given that this appears to be the first paper to demonstrate the effect of an acoustic link and 
acoustic knot, the authors should add some addition emphasis of their analysis and results in context 
of previous results in related fields, such as optics. In particular, the authors should comment on if 
there are any differences between their equations and measured results for acoustic (longitudinal) 
waves and previous results with electromagnetic (transverse) waves. 
 
2) While the Methods section is overall very detailed, the authors should clarify if the metamaterial 
was 3D printed as one entire piece, or was printed in smaller sections and assembled. If it was printed 
in small sections, the authors should describe how they attached those different sections together. 
 
3) A few minor typos were found that should be corrected: 
- In the following text from p. 4: “For example, the transverse size of the sample may be up to 
6mx6m for an audible frequency 6 kHz, suppose that the pixel size is of 1/3 operating wavelength. (ii) 
Except the fabrication trouble of bulky samples…” 
Here, “suppose” should be replaced with a different word, such as “given”, and “Except the” should be 
“Except for the”. 
- In the last paragraph on p. 4, In “To enable of proposal experimentally feasible”, “enable” should be 
“make”. At the end of the same sentence, “24 multiplied by 24 pixels” should just read “24 x 24 
pixels”. 
- In the first line of p. 6, “At the case” should be “For the case”, 
 
Once the issues listed above are addressed, I recommend that this manuscript is ready for publication 
in Nature Communications. 
 
 
 
Reviewer #3: 
Remarks to the Author: 

In this article, Zhang et al describe the creation of acoustic knots--essentially acoustic pressure 
holograms with knot-like 3D shapes--by means of a planar metamaterial array whose units can 
independently manipulate amplitude and phase. While the metamaterial design is not novel (it was 
introduced by Tian et al. in their 2017 letter "Acoustic holography based on composite metasurface 
with decoupled modulation of phase and amplitude", Ref.[42] in this article), the application to knots 
seems to be. The topic of this article is interesting, and it has the potential to attract the attention of 
others in the field. However, the fact that some concepts are not clearly explained and the absence of 
a powerful demonstration of the usefulness of the phenomenon being investigated prompt me to think 
that this article might be better suited for a technical journal rather than a high impact 
multidisciplinary venue like Nature Communications. 
 
The reasoning behind my statement is elaborated in the following. 
 
1) Concern on novelty. While it is very interesting that the authors were able to create knotted regions 
of negative pressure, I can't help but wonder whether this is something that is easy to achieve given 
the design in Ref.[42]. Can the authors clarify if the novelty of their work goes beyond creating a 3D 
hologram of a knot? Or, how is what they are doing different than the holography demonstrated in 
Ref.[42]? 
 
2) Concern on the applicability of knots for particle manipulation. It seems to me that the biggest 
achievement of this article is the creation of knot-like holograms, where the knots are essentially 
regions of minimum pressure. But is the pressure gradient between high- and low-pressure regions 
enough to concentrate particles at the knot regions? From [Memoli et al, Metamaterial bricks and 
quantization of meta-surfaces, Nature Communications, 2017], particle trapping can occur in low 
amplitude pockets surrounded by high pressure regions. But, in the current work, the low pressure 
regions seem to be distributed and wide, rather than highly localized. Do the authors have any insight 
in this? 
 
3) The article seems scientifically sound, but it lacks a field-opening demonstration that would justify 
publication in a high-profile journal like Nature Communications. Have the authors tried to 
demonstrate that their knots can indeed be used for complex 3D particle manipulation? In my opinion, 
a demonstration of this type would raise the article to a much higher level. Just speculating on the 
potential applicability of these knots, in my opinion, is not enough. 
 
4) About topological protection of the knotted shapes. The authors mention in several points of the 
article that their knots are topologically-protected, and this is the reason why 24x24 cells are enough 
to see knot-like shapes. I found this argument very hand-wavy, and unsatisfactory. How does 
topology play a role here? If you lower the number of cells below 24x24, do you still see knotted 
shapes owing to this topological protection? When would this protection cease to be effective? I urge 
the authors to be clearer, and more rigorous, about this. 
 
5) Level of detail. Most explanations in the article are not very clear; this stems from the fact that the 
authors do not provide a sufficient level of detail in the main article. Sure, the Supplementary Material 
(SM) helps. But the SM should not be used as a place to dump all the explanations, leaving the main 
article to become a superficial treatment of the topic. 
- For example, the authors talk about optimization to design the metamaterial with 24x24 cells. But 
how is this optimization done? Why did the authors choose 24x24 cells? What happens if they choose 
more or less? This is not even clear from the SM. 
- The geometry of a unit cell is described in the caption of Fig.1, and the authors mention the pipeline 
length L in both the main text and caption. However, from the figure, it is not really clear what L is. It 
is also not clear if all that matters is the path length or its shape. 
- Why do the authors operate the metamaterial at a frequency of 6 kHz? There is absolutely no 
explanation on this point in the article. Did they choose the frequency arbitrarily? This demonstrates 
that the treatment is superficial and that the work is hardly reproducible the way it is at the moment. 
- How are the 3D reconstructions of the knots in Fig.3 obtained? Are the authors tracking the minima 
of the pressure slices shown in the same figure? It seems to me that these pressure maps present 
entire regions with pressure close to 0. So, how do the authors select only two or four points per slice? 
Are they local minima? This is absolutely not clear from the text. Also, how is the trefoil knot colored 
in Fig.3b? 
 
Minor suggestions 
 
- The work is not badly written, but the article "the" is incorrectly used in various parts of the text. 
This should be fixed. 
 
- The authors, in the main text, comment on the partial differences between numerical and 
experimental knots. However, these differences are very hard to see, since the two sets of knots are 
not plotted next to each other. I think the authors should add the numerical 3D knots to Fig.3, and 
plot both numerical and experimental knots from an identical viewpoint. Only then, it would be 
possible for the reader to compare them directly.