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Topological interpretation of color exchange invariants: hexagonal lattice on a torus

by O. Cépas, P. M. Akhmetiev

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Submission summary

Authors (as registered SciPost users): Olivier Cépas
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-02-05
Date submitted: 2021-01-25 09:18
Submitted by: Cépas, Olivier
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We explain a correspondence between some invariants in the dynamics of color exchange in a 2d coloring problem, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.

Published as SciPost Phys. 10, 042 (2021)

Author comments upon resubmission

Dear Editor,

We would like to resubmit our manuscript to SciPost Physics. We have very much appreciated all the referee's comments and have changed the manuscript accordingly. More background, clarifications, and definitions have been given for the manuscript to be self-contained and more readable for a non-expert. The title has been changed to be more precise, as requested. We hope that this is now suitable for publication.

Best Regards,
the authors.

List of changes

The title has been changed.

The introduction has been modified. The second sentence now broadly defines "color exchange". The second paragraph mentions the "q-state Potts model" as suggested by a referee. The third paragraph has been partly rewritten to include more background on dimer models (not any longer defined as "two-color" model), and proper definitions. Page 3. Last but one paragraph: more motivation for "dynamics II" is given with some broad references added.

Page 4. Section 2.1. The definitions have been more clearly given and two figures added (fig 1 and 2).

Page 6. Section 2.2. The motivation of "dynamics II" is explained in the first paragraph.

The conclusion has been split into a real conclusion and "perspectives and open problems" (expanded a little).

We have changed the notation for the parity P->R to avoid possible confusions with the polynomial P_5.

The following references were added, to give more background:

C. Castelnovo, P. Pujol, and C. Chamon, Phys. Rev. B \textbf{69}, 104529 (2004),

R. Moessner and K.~S.~Raman, \textit{Highly Frustrated Magnetism}, Eds. C.~Lacroix, P.~Mendels, F.~Mila. Springer Verlag (2010).

V.~I.~Arnol'd, Funct. Anal. Appl., 23:3, 169–177 (1989).

V.~A.~Vassiliev, \textit{Complements of discriminants of smooth maps: topology and applications}, 2-d extended edition, Translations of Math. Monographs, 98, AMS, Providence, RI, 1994, 268 pp.

S.~Fisk, Adv. Math. \textbf{11}, 326 (1973); S.~Fisk, Adv. Math. \textbf{24}, 298 (1977); S. Fisk, Adv. in Math. \textbf{25}, 226 (1977).

O. C\'epas and T. Ziman, Progr. of Theor. Phys. Suppl. 159, 280-291 (2005).

Reports on this Submission

Anonymous Report 3 on 2021-1-29 (Invited Report)


It seems that there is broad agreement on the strengths and weaknesses of the manuscript between the referees, and that the latter have largely been addressed. I therefore recommend acceptance.

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Anonymous Report 2 on 2021-1-27 (Invited Report)


The authors have taken into account the criticisms and suggestions of the referees, and have made the appropriate amendments. In my opinion, the paper is now more accessible to the non-specialist reader and self-contained.
I recommend this paper for publication in SciPost Physics.

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Anonymous Report 1 on 2021-1-27 (Invited Report)


The authors have responded to the referees' comments and have considerably clarified the introduction. I recommend publication.

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