journal of mathematics and the arts

[RobinHankin]

submitted inst/jma_knot.tex to Journal of Mathematics and the Arts.

The submission ID is: 209604076

[RobinHankin]

no email confirmation yet; they say "We have sent confirmation to hankin.robin@gmail.com and will notify you to keep you up to date with the progress of your submission. If you have any queries, please get in touch at journalshelpdesk@taylorandfrancis.com with your submission ID in the subject line"

[RobinHankin]

Dear Customer,

This is an automatic email response.

Thank you for contacting the Taylor & Francis Authors Helpdesk. We will endeavour to reply to your email within 2 working days. I emailed them and they replied:

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[RobinHankin]

they write back on 11 november2020

RE: manuscript 209604076 [ ref:_00D0Y35Iji._5007R2o4KnN:ref ]

Dear Robin,

Thank you for your email.

I have checked the status of your manuscript within our online submission system and I can confirm that your submission is currently progressing through peer review without any problems.

I am unable to provide any more specific information regarding your submission at this time, aside from assuring you that we will do our best to deliver a decision to you as soon as possible.

Please do not hesitate to contact me if you have any further questions or concerns.

[RobinHankin]

just a note that in February 2021 I misread the journal page and thought that the manuscript was not entered correctly. So on 11 Feb 2021, I submitted a new manuscript, number 217578667. But this was a mistake and I withdrew it on 12 Feb 2021.

[RobinHankin]

uploaded revision to JMA. Automatic email response:

Thank you for submitting your revised manuscript.

Submission ID 209604076 Manuscript Title
Visually pleasing knot projections Journal Journal of Mathematics and the Arts

You can check the progress of your submission, and make any requested revisions, on the Author Portal.

Thank you for submitting your work to our journal. If you have any queries, please get in touch with journalshelpdesk@taylorandfrancis.com.

Kind Regards, Journal of Mathematics and the Arts Editorial Office

[RobinHankin]

Email from the editor, 30 May 2022:

Comments from the Editors and Reviewers:

Thank you for this revised manuscript. My apologies for contributing to the slow turnaround of these reviews.

I have given this version the official status of "Minor Revision" to reflect the fact that we all agree that this draft is much closer to publication-ready. In particular, I believe that the remaining reviewer concerns can be addressed with moderate rewriting. However, some of the reviewer feedback falls closer to "Major Revision," so I wanted to acknowledge that a few of the remaining critiques are beyond surface level.

As you will see in the detailed feedback below, one of the reviewers still has substantive concerns about the overall framing of your discussion and how directly it addresses the mathematical and aesthetic choices that went into the software package. I suspect that you can address the concerns in the first few review paragraphs by being more specific about the constraints of your implementation, the level of flexibility you hoped to provide for the user, and the alternate possibilities for future work, but since I don't know what technical decisions went into your coding, I might be misjudging exactly what is suitable.

I look forward to receiving your further revision, and will make sure the editorial process goes more quickly from this point forward.

Best wishes,

Susan Goldstine, Ph.D. Associate Editor Journal of Mathematics and the Arts

Reviewer #1: The paper has improved in terms of its overall presentation in the context of knot renderings, and many detailed fixes have been made as suggested by the reviewers. However, the paper still has the major weakness of being too much a somewhat ad-hoc technical manual on "How-to-use-R and Inkplot to get some decent knot plots" by using the software package [27]. The paper is lacking good discussions of the key issues and explanations of the choices made, specifically: What makes a knot plot attractive, followed by a thorough discussion of how to best set up a knot representation and an optimization procedure to achieve the desired qualities?

The problem starts with the basic curve representation: Why use Bezier curves? Why not use some other spline curve that guarantees G2-continuity? And, if forced by the chosen tool to use Bezier curves, why not place nodes at the crossing points, (and one or two more for large loopy lobes between subsequent crossing points)? -- This would allow the user to place the crossing points far enough apart to start with, and this would make it easy to set the Bezier handles to force right-angle crossings. Also, the implementation in the "badness function" seems far from optimal. "Total crossing angles()" does not address a key problem directly: If one wants to avoid highly acute crossing angles, then this can be addressed in a much more sensitive way with a function such as 1/ sqr( cross.angle ). This would go to infinity in the worst case! It also corresponds more directly to the way that excessive curvature is avoided by minimizing "bending energy," which is the integral over the square of curvature -- or: 1/ sqr( curve-radius). These are some "math" issues that readers of JMA would be interested in.

Some more detailed comments:

Page 3 line 42: "…knots possess a line of symmetry (at least, the diagrams do if the breaks are ignored), which …" ->> '…knots possess a C2 rotation axis, which …'

Page 4 lines 22-25: "One plausible technique for creating knot projections is to consider a two-dimensional projection of a knot's embedding E in R3. However, this approach often results in poor visual appearance: cusps or other displeasing effects can occur." ->> True! - But what is the alternative, if the user has an entangled physical knot in her hands?

Page 4 line 45: Define "nodes" !

Page 4 line 48: "However, if we can ensure that the radius of curvature changes only by a small amount,…" ->> HOW can the user do this?

Page 4 lines 50-52: No need to differentiate here between over- and under-strands. I assume they are treated the same in the optimization process.

Page 4 lines 53-54: "We would like strand crossing points to be far from nodes,…" -- Why? The curves are also visually continuous across the nodes!

Page 5 lines 46-51: Why are there SEVEN significant decimal digits? Does the graphic screen have 10 million pixels per line?

Figure 3, 5, 7: Perhaps fewer than 100 circles per Bezier curve would be better.

Figure 4. Knot 76 with strands numbered… -- The main use here seems to be to see very clearly the extent of every Bezier curve. Over-/under-crossings seem to become relevant only for the final rendering of the knot curve with the gaps included.

Page 7 line 52: "The optimization typically proceeds over R 50" - What is the meaning of this?

Page 7 line 56: "…function total crossing angles()" The function is not explicitly defined in the paper, but it sounds clearly suboptimal. One bad acute-angle crossing can easily "hide" in a sum of all angles. It would be better to individually penalize acute angles. A suitable term might be: 1/ sqr( cross.angle ). This would go to infinity in the worst case!

Page 8 line 20, and footnote: The "devil" and the "interesting math" is in the details! The footnote is not sufficient to explain the trade-offs that are being made in the optimization process.

Page 9, line 62: What is "Bosch-type symmetry:…" ? Please explain.

If all nodes fulfill the specified symmetry constraints (as shown on page 11, lines 15-22 and on lines 35-43), but the connection diagram does not completely adhere to the specified symmetry - What happens? - Does the package check this and let the user know about the discrepancy?

Page 10 lines 56-58: "Minimizing the badness is not entirely straightforward…" -- What is the problem? I think that if all the points are moved jointly within the specified symmetry constraints, an optimized symmetrical solution will result.

{The above two comments suggest, that it might be advantageous to use some hierarchy in the representation of symmetrical knots: Only specify the curve for one unique fragment of the knot curve, and compose the complete curve by instancing that fragment with mirroring and rotation as required to construct the complete knot! The optimization would then only have to be done for this unique fragment (with proper end-conditions). If this same hierarchy is also used in the graphical display of the knot curve, the user could easily make adjustments to just a few nodes, while always automatically obtaining the desired symmetrical result.}

Page 10 lines 55: "In the package, the badness weightings may be altered easily,…" -- What is the user interface for this? Are there sliders in the display that allow interactive adjustments of the weights?

Figure 9: The fact that "angle-crossing penalty" had to be enhanced by a factor of 100 to achieve a more desirable (in my opinion) result, illustrates the fact that the "total crossing angles()"-function is not a good choice for optimization.

I am missing a description how the two breaks at every crossing point are being graphically implemented.

Conclusions: RE: "Further work might include functionality to deal with links."

Readers may think that this would be a trivial extension. Please add a paragraph like the one you wrote in response to Reviewer 1, to explain why the extension to links may be more difficult than one might think.

RE: "In the broader context of optimization in art, we observe that numerical optimization techniques can produce aesthetically pleasing results, an observation that might find uptake by graphic artists."

This only holds with the proviso that we can think of a good mathematical formulations to capture the various features that capture aesthetics appropriately. - This then becomes the real "art."

Gallery: Knot diagrams 83 through 86 in Figure 12 stand out with their helices too tightly wound. The two criteria on page 4, lines 54 and 55 do not seem to be sufficiently enforced. I suggest adjusting these figures in the spirit of Figure 9b. This may happen automatically with a better "badness function" that penalizes non-orthogonal crossing angles more directly.

[27]: Incomplete reference. Perhaps give a URL.

Reviewer #2: Firstly I would like to apologise for my late response on this. Reviewing this edited paper had not got onto my radar.

Secondly all the points in my original report have now been addressed. I do feel that the literature review could have gone a little bit further beyond my stated suggestions, but it is now sufficient to give the paper more context.

[RobinHankin]

Email from the editor, Mara Alagic, 30 January 2023:

Comments from the Editors and Reviewers:

I appreciate and accept most of the comments made by the author in his rebuttal and the corresponding fixes in the manuscript. But I have a remaining concern that much of this material is perfect for an on-line tutorial on how to use package [27] to draw optimized knot diagrams, but JMA is not the right place to publish such a tutorial.

The content that is most fitting for a JMA paper is a discussion of what makes a knot diagram "beautiful" - and that discussion will clearly be biased by the personal view of the author (perhaps enhanced by a "user-study" among family and colleagues). Thus lines 14 - 20 on page 4 are of prime importance and could be fleshed out more. The main contribution to JMA would then be a detailed discussion of how the author has struggled to capture these goals in the form of mathematical functions that can be optimized and will then approximate these goals. In this respect, the manuscript still has some shortcomings. They may be fixed in two ways: Enhance the detailed explanations of the choices made in creating package [27]; and add a more extended discussion of a few things that might be approached differently in a future package ( -- which can also handle links). In view of this, I recommend that Appendices A and B should be firmly integrated into the main text. The discussions therein relate rather directly to what I see as the main contribution of a JMA paper on this topic.

A few comments on some rebuttals:

RE: Why use Bezier curves? : A key point, which should be mentioned in the justification for Bezier curves, is that they interpolate their end-points. This gives users a convenient, more direct control of where these curve segments will appear than what they would get from G2-continuous B-splines, which are only approximating their control points.

RE: Why not place nodes at the crossing points? : Yes, this will require a different, slightly more sophisticated drawing interface; but this may then pay off heavily in a simpler, more efficient optimization process. For each crossing point, the user would see an orthogonal "cross-hair" that can be shifted and rotated as a whole. In addition, there are two parameters (handle-pairs) that control the "velocities" of the two strands through the crossing. The optimization routine would fine-tune these velocities to achieve matching curvature for the two sequentially connected Bezier curves. And, yes, if a crossing point falls on a line of mirror symmetry, the crosshair would automatically be angled at 45 degrees with respect to this line. And if the knot has n-fold rotational symmetry, n such cross hairs would be placed at the vertices of a regular n-gon. (All this is not such a "nightmare" if it is properly considered from the beginning.) I don't expect that anything in the current system should be changed; but this approach should definitely be considered in a future version of such a package.

RE: Mismatched diagram and symmetry specifications: "This situation cannot occur" -- If the user makes a mistake, will the current system send a suitable error message?

RE: A hierarchical way of dealing with symmetry: "This is exactly what I have done! The referee gives an accurate summary of the code's methodology."

• Place a corresponding, extended description into the main text.

RE: A slider interface for changing some optimization parameters: It is clear why this would not be practical in the current system, where the optimization of a pretzel knot "may take a few weeks." But this also suggests that there is much room for improvement! Perhaps some badness functions are not really needed or are badly implemented.

RE: Link projections: In a future system, the planning for handling links should come in right at the beginning. It is not at all clear why "Each component would have its own badness." Many of the important badness functions (e.g. separations of nodes or of non-crossing curve segments) are based purely on local geometry and need not know what component of a link they refer to. Perhaps the overall length of a knot or of a link-component is not a good metric, and should be replaced with some other functions of the length of curve segments between crossings.

Some detailed comments on manuscript R2:

Page 6, line 20: How many (x,y) pairs are there? Page 6, Line 36: Why "R50"? - Are there a total of 50 parameters? What are they? Page 6, Line 39: "… the overall bending energy." ->> 'minimizing overall bending energy.' Page 6, Line 42: What is "> b <" ? Figures 3 and 5: Try running these figures with only 50 points (and circles) for each Bezier segment. Figure 4 caption: Much of the explanation of "7 3" versus "3 7" is already in the main text. Page 8, Line 60: "Why 64 real variables"? What are they? How does this agree with "R50" on page 6 ? Figure 7: Where does the spiky artifact in the upper right lobe come from? Page 12, Line 52: "… a rotationally symmetric" ->> " … a knot with pure cyclic rotational symmetry, rather than dihedral symmetry" Figure 9: Why did the user choose to place three nodes on the outer lobes, rather than only two? - Was this easier to express the desired symmetry??

Page 16, Line 47: "The strands exert a couple but no force on one another." ?? Lengthy discussions should be relegated to the main text, rather than be in figure captions.

Figure 14: All the knots are nicer then the ones in Ref [24]. Good results!

References: Several of them are incomplete and need an URL or a more detailed description where the source can be found.

[RobinHankin]

uploaded this version to the JMA editorial softare portal just now.

[RobinHankin]

Dear Robin Hankin,

Thank you for submitting your revised manuscript.

(automated) reply from JMA:

Submission ID 209604076 Manuscript Title
Visually pleasing knot projections Journal Journal of Mathematics and the Arts

You can check the progress of your submission, and make any requested revisions, on the Author Portal.

Thank you for submitting your work to our journal. If you have any queries, please get in touch with journalshelpdesk@taylorandfrancis.com.

Kind Regards, Journal of Mathematics and the Arts Editorial Office

[RobinHankin]

Feb 13, 2023

Ref.: Ms. No. TMAA-2020-0042R3 Visually pleasing knot projections Journal of Mathematics and the Arts

Dear Robin Hankin,

Reviewers have now commented on your paper. I am pleased say that I would like to accept your manuscript subject to you making some very minor revisions. If you are prepared to undertake the work required, I would be pleased to publish your paper in Journal of Mathematics and the Arts.

1. The manuscript has to be in an editable format (Word or Latex)
2. All images/figures have to be submitted as separate files, high quality! (https://www.tandfonline.com/action/authorSubmission?show=instructions&journalCode=tmaa20)

Your revision is due by 02/26/2023.

To submit a revision, go to https://rp.tandfonline.com/submission/flow?submissionId=209604076&step=1. If you decide to revise the work, please submit a list of changes or a rebuttal against each point which is being raised when you submit the revised manuscript.

If you have any questions or technical issues, please contact the journal's editorial office at SP-ingest@journals.tandf.co.uk.

Yours sincerely,

Mara Alagic, Ph.D. Editor Journal of Mathematics and the Arts

[RobinHankin]

Ref.: Ms. No. TMAA-2020-0042R4 Visually pleasing knot projections Journal of Mathematics and the Arts

Dear Robin Hankin,

I am pleased to tell you that your work has now been accepted for publication in Journal of Mathematics and the Arts.

It was accepted on Feb 23, 2023

Comments from the Editor and Reviewers can be found below.

Thank you for submitting your work to this journal.

With kind regards,

Mara Alagic, Ph.D. Editor Journal of Mathematics and the Arts

[RobinHankin]

Any copyrighted material reproduced in your paper must include an accompanying attribution. Brief extracts of third-party material may be cleared for use under the fair use / fair dealing policy, and don't require full copyright clearance from the Rightsholder. For further information and to access a template form for requesting permission, please see https://authorservices.taylorandfrancis.com/using-third-party-material-in-your-article/. Please keep copies of all correspondence.

Article: Visually pleasing knot projections

Journal: Journal of Mathematics and the Arts JMA

Article ID: JMA 2185058

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[RobinHankin]

Published!

Robin K. S. Hankin (2023): Visually pleasing knot projections, Journal of Mathematics and the Arts, DOI: 10.1080/17513472.2023.2185058 To link to this article: https://doi.org/10.1080/17513472.2023.2185058

[leaving open until I sort out citation info in the package]