Pith. sign in

REVIEW 3 major objections 5 minor 64 references

Geometric Frustration in Twist-Bend Nematic Droplets

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Spherical confinement plus radial anchoring turns achiral twist-bend nematics into a menagerie of metastable textures, including a numerically stable Hopfion in a narrow parameter window.

desk verdict A genuinely useful catalogue of twist-bend droplet textures, with a Hopfion stability claim that is honestly flagged but not yet nailed down. read the letter →

arxiv 2505.16140 v1 pith:JQ55ZYVB submitted 2025-05-22 cond-mat.soft

Joseph Pollard , Richard G. Morris This is my paper · ORCID
classification cond-mat.soft MSC 82D30 PACS 61.30.-v61.30.Jf
keywords twist-bendnematicgeometricfrustrationHopfionliquidcrystaldropletsradialanchoringpseudolayersdefecttextures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Twist-bend nematics are liquid crystals made from bent-core molecules that prefer a nonzero bend distortion, a preference they relieve by twisting into a heliconical structure even though the molecules are achiral. This paper asks what happens when such a material is confined to a spherical droplet with radial anchoring, a geometry already known to force cholesteric droplets into elaborate metastable textures. Using numerical energy minimisation of a polarisation-field model, the authors map the equilibrium textures as the molecular cone angle and the pitch-to-radius ratio vary, and they compare each texture with its cholesteric counterpart. The central result is that twist-bend droplets reproduce many cholesteric-like layered states and defect constellations with different local geometry, and that a Hopfion, a linked pair of soliton tubes, is numerically stable in a narrow parameter window despite the material being achiral. The catalogue matters because it predicts which textures should appear in experiments and singles out Hopfions and torons as rare or transient.

What carries the argument

The argument is carried by the polarisation-field free energy $E=\int \left(\frac{K}{2}|\nabla \mathbf{n}|^2+\frac{C}{2}|\nabla \mathbf{p}|^2-\lambda \mathbf{p}\cdot \mathbf{b}+\frac{U}{4}(1-|\mathbf{p}|^2)^2\right)$, in which the auxiliary polarisation $\mathbf{p}$ represents the bent-core molecular shape and couples to the director bend $\mathbf{b}$, setting a preferred nonzero bend. Its bulk ground state is the heliconical director $\mathbf{n}_h=\cos\theta_0\,\mathbf{e}_z+\sin\theta_0(\cos qz\,\mathbf{e}_x+\sin qz\,\mathbf{e}_y)$, with cone angle $\theta_0=\pi/M$ and inverse pitch $q=2N\pi/R$ leaving two control parameters. To classify the minimised textures the authors use Pontryagin–Thom surfaces to visualise pseudolayers, Morse indices to label point defects, linking of director preimages to compute the Hopf invariant, and $\beta$-lines, zeros of the bend vector, as the twist-bend analogue of the $\lambda$-lines of cholesterics. This combination converts the known phenomenology of cholesteric droplets into twist-bend language and separates topologically required features from purely geometric ones.

What would settle it

Re-run the energy minimisation for $M=5$, $N=6$ to $10$ and $M=5.5$, $N=10$ to $12$ on finer grids, for example $200^3$ and $300^3$ points, with a stated energy-convergence threshold, and check whether the two linked preimages $\mathbf{n}=\pm \mathbf{e}_x$ remain linked after continued relaxation; if the Hopfion always decays to a toron or bowl texture on finer grids, the claimed stability window is a numerical artifact.

Watch

Extended reading notes

Core claim

The paper claims that a twist-bend nematic confined to a radially anchored spherical droplet supports a wide family of metastable equilibrium textures: twisted hedgehogs, twisted hyperbolic defects, Mexican-hat and pinwheel pseudolayers, bowl, onion, screw, flower and Frank–Pryce-type structures, defect strings, torons, and Hopfions. Stability is controlled by the cone angle $\theta_0=\pi/M$ and the dimensionless inverse pitch $N=qR/2\pi$, with $N\in[1,12]$ and $M\in[5,12]$. Many of the structures are topologically analogous to cholesteric droplet textures but geometrically distinct, because the twist-bend material has no preference for one handedness and because its bend field can vanish on networks of $\beta$-lines that are not topologically required. The most specific claim is that a Hopfion, identified by linked preimages of the director, is numerically stable for roughly $N=6$ to $10$ at $M=5$ and $N=10$ to $12$ at $M=5.5$, in an achiral material where the Derrick–Hobart theorem forbids Hopfions in the ordinary nematic case. Quenches from the isotropic phase never produced a Hopfion in 100 runs per parameter set, so such solitons are expected to appear with frequency below about one percent.

Load-bearing premise

The load-bearing premise is that the numerically minimised textures, especially the Hopfion in its narrow parameter window, are genuine equilibrium states of the continuum free energy rather than artifacts of the finite grid or the stopping threshold.

Editorial extensions

If this is right

  • Twist-bend droplets with radial anchoring should display flattened, screw-like, cylindrical and spherical pseudolayers that resemble cholesteric droplet textures, but with U-shaped or tightly coiled director integral curves and dense networks of $\beta$-lines.
  • High-charge point defects of charge $-2$ and $-3$ are not stable in the surveyed parameter range; they split into strings of $-1$ defects, because boundary hedgehogs are not pinned as firmly as in cholesterics and reversed-handedness regions cost less energy.
  • Hopfions and torons should be observable only as rare or transient structures; in 100 quenches per parameter set no Hopfion appeared, placing their quench frequency below roughly one percent.
  • Realistic droplet radii for these textures are 10–120 nm given the roughly 10 nm pitch of twist-bend phases, and the regime includes the material CB(CH2)7CB with pitch 8.3 nm and cone angle about 24 degrees.
  • Larger cone angles and larger values of $N$ than in cholesterics are generally needed to stabilise the more complex layered textures, so experiments should search for the most intricate structures in small droplets of strongly bent-core materials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the paper uses equal elastic constants while real twist-bend materials have a notably smaller twist constant, the narrow Hopfion window is likely a conservative estimate; a realistic anisotropic model, which the paper notes could enlarge it, may make Hopfions easier to stabilise experimentally.
  • A testable extension is to initialise minimisations from hyperbolic-space pure-bend textures, such as horocycles in the Poincaré ball, rather than cholesteric Lyre textures; if the Hopfion window widens, the stabilising mechanism is the bend geometry itself rather than the chosen initial condition.
  • If Hopfions are confirmed experimentally, their absence in isotropic quenches implies that they must be prepared deliberately, for example by optical or flow-based methods akin to those used in cholesterics, so their rarity would be an assembly challenge rather than a stability issue.
  • The prevalence of excess $\beta$-lines that are not topologically required suggests a measurable energy cost: comparing a texture with its $β$-line count removed by local reconnection could test whether geometric frustration, rather than topology, controls which metastable state is selected.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies spherical droplets of a twist-bend nematic liquid crystal with radial anchoring by numerically minimizing a continuum free energy that couples the director to a polarisation-like field (Eq. (1)). The two control parameters are the cone angle θ0 = π/M and the pitch-to-radius ratio encoded in N. The authors catalogue many equilibrium and metastable textures (twisted hedgehog, twisted hyperbolic defect, Mexican hat, pinwheel, bowl, onion, screw, flower, Frank–Pryce, defect strings) and compare them with cholesteric-droplet analogues. The central novelty is a reported narrow window of numerically stable Hopfions (M = 5, N ≈ 6–10; M = 5.5, N ≈ 10–12), obtained from a cholesteric Lyre initial condition (Eq. (13)), together with a toron decay channel and quench statistics showing no Hopfions in 100 runs per parameter set.

Significance. If the Hopfion claim is correct, the paper makes a notable contribution: it suggests that achiral twist-bend nematics under radial confinement can host Hopfions, and it substantially extends the known analogy between cholesteric and twist-bend droplet textures. The paper is also careful to use topological diagnostics (preimage linking, Morse indices, β-lines, Pontryagin–Thom surfaces) and to state many of its limitations explicitly. The specific parameter window is a falsifiable prediction that could guide experiments. The strength of the paper is therefore real but contingent: the Hopfion stability claim, which is the most novel and load-bearing assertion, is not yet supported by the numerical evidence presented.

major comments (3)
  1. [Section III; Appendix D] The central claim of numerically stable Hopfions is not yet established. Section III states that Hopfion behaviour is "quite sensitive to numerical parameters such as the number of points used in the discretisation" and that "Hopfions correspond either to a very narrow minimum in the energy, or else lie close to a slow trajectory past a saddle." Appendix D gives neither the actual energy-change threshold used to terminate minimizations nor a Hopfion-specific grid-convergence study; the general statement that 150^3 and 200^3 grids made no substantial difference is not a substitute, because Hopfion identification depends on linking of two preimage tubes (Appendix B) and an under-resolved grid can preserve linking artificially. I request a Hopfion-specific convergence study for the claimed windows, the numerical tolerance value, and evidence that the final states are true local minima, for example an extended energy plateau and decay of small superimposed perturbations.
  2. [Section I; Eq. (1); Section III] The one-constant approximation is admitted in Section I to be "not really valid" for real twist-bend materials, and Section III notes that elastic anisotropy affects Hopfion stability in cholesterics. Because the reported Hopfion window is narrow, this approximation is not innocuous for the central claim: the equal-constant model may either create or destroy the Hopfion window. A concrete test would be to repeat the M = 5, N = 6–10 and M = 5.5, N = 10–12 minimizations with elastic anisotropy representative of a known material such as CB(CH2)7CB, which the paper itself uses in Section IV. Until such a test is performed, the abstract and conclusions should present Hopfion stability as a one-constant-model prediction rather than as an unconditional property of twist-bend droplets.
  3. [Section III; Section IV] The only reported route to Hopfions is the cholesteric-adapted Lyre initial condition of Eq. (13), and the quench statistics in Section IV found zero Hopfions in 100 realizations per parameter set. This absence is consistent with a small basin of attraction, but combined with the paper's own saddle hypothesis it leaves open the possibility that the Hopfion textures are long-lived transients rather than equilibrium states. I ask for an independent stabilization test, for example starting from the converged Hopfion, applying random director perturbations of controlled amplitude, and verifying return to the same linked-preimage state, or computing a Hessian/Morse-index indicator. This would separate "rare but stable" from "kinetic artifact".
minor comments (5)
  1. [Eq. (7)] The radial hedgehog is written as n = (x ex + y ex + z ex)/r; the y and z components should use ey and ez.
  2. [Introduction] The phrase "they exhibit stabile chiral structures" contains a typo; "stabile" should be "stable".
  3. [Section I, after Eq. (3)] The sentence "relate the parameter ratios ... to to the cone angle" contains a duplicated "to".
  4. [Fig. 8 caption] The caption reads "M− 5" where it should read "M = 5".
  5. [Appendix D] The grid is described as "1003 grid points"; this should be written as 100^3 to match the later notation 150^3 and 200^3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the parameter map in Eq. (3) is a bulk reparametrization, and the Hopfion stability window is a genuine numerical output that fails for most parameters.

full rationale

The derivation chain is a numerical exploration rather than an analytic derivation, and I find no step where an output is built into an input. The mapping in Eq. (3) takes the bulk heliconical ansatz, substitutes it into Eq. (1), and minimizes with respect to theta0 and q; it is a reparametrization of the simulation parameters (lambda/K, C/K) into the more physical (theta0, q, N, M), and no droplet observable is used to determine these constants. The Hopfion result in Section III is obtained by minimizing the twist-bend energy from the cholesteric Lyre initial condition Eq. (13); because the same initial condition decays to torons or bowl textures for most parameters (e.g., M=6, N=10, M=7, and large N) and survives only in the reported window, the stabilized Hopfion is a numerical output rather than a restatement of the initial condition. The self-citations (Refs. [4], [21], [27]) supply definitions and tools such as beta-lines, Morse indices, and the Euler-class formula (B1), but these are either standard mathematics with external correlates (Refs. [31], [46], [47]) or descriptive vocabulary, and they do not fix the equilibrium textures. The paper's own caveat that Hopfions may be a very narrow minimum or a slow trajectory past a saddle is a numerical-convergence caveat, not evidence of a fitted circular step. Consequently, the central catalogue is self-contained computational evidence, and no circularity is present.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. beta-lines and pseudolayers are descriptive/geometric tools taken from Ref. [4], and Hopfions and torons are existing topological soliton concepts studied previously in cholesterics. The free parameters listed are either insensitive scale choices or initial-condition parameters stated to be non-essential.

free parameters (3)
  • U (polarization length constraint stiffness) = 0.1
    Hand-chosen; authors state varying U in [0.01,1] made no real difference to equilibrium structures, so it is a scale parameter rather than a fitted one.
  • K (elastic constant) = 0.1
    Sets overall energy scale and is the same for all simulations; absorbed by rescaling and does not affect reported textures.
  • Lyre initialization parameters rho0, tau0 = rho0=0.6R, tau0=0.2R
    Hand-chosen for the Hopfion initial condition; the paper states 'this choice is not important for the equilibrium structure'.
assumptions (6)
  • domain assumption Meyer/Kent State free energy Eq. (1) with a director field and auxiliary polarization p correctly describes the twist-bend nematic phase.
    Adopted from Refs. [1-3, 23-25]; the paper does not derive this from a molecular model and it restricts the defect types to point defects and solitons.
  • domain assumption One-constant elastic approximation (single K for all director distortions) is sufficient to reproduce qualitative equilibrium structures.
    Stated in Section I as 'not really valid for these materials', justified by analogy to cholesteric simulations; it affects the quantitative parameter windows, including Hopfion stability.
  • standard math The heliconical ansatz Eq. (2) is the bulk ground state, and Eq. (3) maps lambda/K and C/K to cone angle theta0 and inverse pitch q.
    Derivation cited to Ref. [4] (co-authored by Pollard); the minimization is parameter-free but not reproduced here, and it restricts cone angles to [0, pi/4).
  • domain assumption Director field representation precludes disclination lines, so all reported textures are in the class of point defects and solitons only.
    Explicitly noted in Section I; real TBN materials exhibit disclinations, so the catalogue may be incomplete.
  • standard math Topological constraints: Poincare-Hopf theorem, Euler class formula Eq. (B1), Hopf invariant from preimage linking, and Morse index classification.
    Used for interpreting textures and calculating charges in Appendices A, B, C; these are established mathematical results.
  • domain assumption Large U effectively enforces unit length of the polarization field.
    Used to derive Eq. (3); with the chosen U=0.1 the constraint is approximate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometric Frustration in Twist-Bend Nematic Droplets." pith.science (2026). https://pith.science/paper/JQ55ZYVB

@misc{pith2026250516140,
  author       = {Pith},
  title        = {Pith review of: Geometric Frustration in Twist-Bend Nematic Droplets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQ55ZYVB}},
  note         = {Machine review of arXiv:2505.16140}
}
read the original abstract

Liquid crystals formed of bent-core molecules are exotic materials that exhibit the twist-bend nematic phase. This arises when an energetic preference for nonzero local bend distortion is accommodated via twist in the texture, resulting in properties synonymous with both smectics and cholesterics. Here we describe how the frustration inherent to the twist-bend phase can be exacerbated by confinement and boundary anchoring. Using a combination of numerical simulations, topological and geometric analysis, we catalogue the equilibrium textures that arise in spherical twist-bend droplets with a radial anchoring as the two key parameters -- the molecular cone angle and the ratio between the pitch length and droplet radius -- are varied. This form of confinement is known to produce a wide variety of topologically and geometrically complex metastable states in cholesterics. We find that twist-bend nematic droplets are no different, exhibiting a variety of complex layered states, defect constellations, and Hopfions. However, whilst many of the structures and defect configurations that we observe are equivalent to their cholesteric counterparts, they are geometrically very distinct, in part due to the absence of chirality.

Figures

Figures reproduced from arXiv: 2505.16140 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic diagram of a bent-core molecule (green) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Structures containing a central point defect. (a) In a cholesteric at [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Structures containing a boundary point defect. (a-c) At [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The negatively curved hyperbolic space [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Structures with flat layers. (a,b) A cholesteric at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Structures with cylindrical layers. (a,b) Cylindrical layers in a cholesteric droplet are the result of inserting a double [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The Frank–Pryce, or RSS, texture. (a,b) This texture is common in cholesteric droplets. Shown here is [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Structures containing strings of defects. (a) A stable defect string in a cholesteric droplet at [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The Lyre structure. (a) The classical Lyre structure in a cholesteric droplet, at [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Equilibrium structures resulting from a quench in a twist-bend nematic at [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 57 canonical work pages

  1. [1]

    In a droplet of radius R we choose q0 = 2N π/R for various values of N —care should be taken when com- paring with other studies, for example Refs

    (5) Cholesterics have a single parameter, q0 = 2 N π/R, which sets the cholesteric inverse pitch length, and which is naturally analogous to the parameter q for twist-bend nematics. In a droplet of radius R we choose q0 = 2N π/R for various values of N —care should be taken when com- paring with other studies, for example Refs. [19–21], where the paramete...

    work page

  2. [2]

    First consider a large cone angle, M = 5

    Twisted Hedgehog Defect In a twist-bend nematic at N = 0—in which case, the director and polarisation are decoupled—the radial hedgehog is stable and remains the equilibrium configu- ration for any value of the cone angle. First consider a large cone angle, M = 5. As we increase N above zero the structure begins to change. While it remains a hedge- hog wi...

    work page

  3. [3]

    However, the equilibrium structure in a twist-bend nematic at N = 4, Fig

    Twisted Hyperbolic Defect For M = 5 and larger values of N ≥ 3, the bending is strong enough to induce a Hopf bifurcation in the cross- section [27, 31], and the defect undergoes a conversion from a hedgehog, MI 0, to a hyperbolic defect of the same charge, MI 2, as it would in a cholesteric. However, the equilibrium structure in a twist-bend nematic at N...

    work page

  4. [4]

    The pseudolayers nucleate along the central β-line and spread throughout the droplet, as shown in Supplemental Movie 3

    Mexican Hat Texture At M = 5 and very large N the structure changes sub- stantially, generating nested families of pseudolayers with a complex structure. The pseudolayers nucleate along the central β-line and spread throughout the droplet, as shown in Supplemental Movie 3. We show the equilib- rium texture for N = 12 in Fig. 2(d). The integral curves of t...

    work page

  5. [5]

    The layers form a ‘pinwheel structure’, and the integral curves of the director are helices with multiple, less dramatic kinks than those observed atM =

    Pinwheel Texture At larger values of N = 6 − 12 and M = 6 − 12 , the central defect again undergoes a conversion to MI 2 and the droplet again fills with pseudolayers, although these have a very different configuration to those that occur at small M . The layers form a ‘pinwheel structure’, and the integral curves of the director are helices with multiple...

    work page

  6. [6]

    We show this for N = 12, M= 11 in Fig. 2(f). The magnitude of the bend is shown on a slice in Fig. 2(g). In these textures, and for the rest of the structures shown throughout, the structure of the polarisation field is similar to that of the bend. However, while the bend is constrained to be orthogonal to the director field the polarisation is not, and t...

    work page

  7. [7]

    Boundary Hedgehogs, the Bowl and Onion Textures We can initialise a director field with a boundary hedgehog (MI 0 defect) by taking the director to be n = ez in the interior and imposing radial boundary conditions. In a cholesteric, the defect remains a hedge- hog close to the boundary for small N , but for N ≥ 2 the defect moves back into the interior an...

    work page

  8. [8]

    (8) In a cholesteric, Fig

    Flat Layers, the Octopus and Screw Textures We next consider flat layers, initialised by the director field n = cos(N πz/R)ex + sin(N πz/R)ey. (8) In a cholesteric, Fig. 5(a,b), the layers (level sets of z) remain flat in the interior of the droplet. The defect (MI

    work page

Show all 64 references
  1. [9]

    Cholesteric λ-lines are dislocations in the layer structure, and these are required to match on to the boundary condition

    appears on the x-axis, and two λ-lines spiral out from it and towards opposite poles on the sphere. Cholesteric λ-lines are dislocations in the layer structure, and these are required to match on to the boundary condition. In a twist-bend nematic at cone angleM = 5 and small N...

  2. [10]

    Cylindrical Layers, The Flower Texture Cylindrical layers are initialised using the director n = − cos(2N πr/R)ez + sin(2N πr/R)eρ, (9) where ρ = p x2 + y2. This initial condition is ‘escape down’ along the z-axis, necessary to produce a right- handed texture [22], but in a tw...

  3. [11]

    This line singularity splits into a pair of +1 singular lines, which then ‘escape’ to form a pair of λ-lines and no director defects

    Spherical Layers, the Frank–Pryce Texture An alternative structure in a cholesteric in a droplet with planar anchoring has spherical layers with a full line singularity of winding +2 connecting the boundary to the centre of the droplet. This line singularity splits into a pair...

  4. [12]

    In a twist-bend nematic these strings are also stable for intermediate N

    The three boundary defects are hedgehogs, MI 0, and the resulting structure forms a V-shaped string of five de- fects which is stable in a cholesteric [21]. In a twist-bend nematic these strings are also stable for intermediate N . A complex network of deformed β-lines connect...

  5. [13]

    Dozov, Europhysics Letters (EPL) 56, 247 (2001), ISSN 0295-5075, 1286-4854, URL https://iopscience

    I. Dozov, Europhysics Letters (EPL) 56, 247 (2001), ISSN 0295-5075, 1286-4854, URL https://iopscience. iop.org/article/10.1209/epl/i2001-00513-x

    2001 doi

  6. [14]

    S. M. Shamid, S. Dhakal, and J. V. Selinger, Phys- ical Review E 87, 052503 (2013), ISSN 1539-3755, 1550-2376, URL https://link.aps.org/doi/10.1103/ PhysRevE.87.052503

    2013

  7. [15]

    J´ akli, O

    A. J´ akli, O. D. Lavrentovich, and J. V. Selinger, Reviews of Modern Physics 90, 045004 (2018), ISSN 0034-6861, 1539-0756, URL https://link.aps.org/doi/10.1103/ RevModPhys.90.045004

    2018

  8. [16]

    Binysh, J

    J. Binysh, J. Pollard, and G. P. Alexander, Physical Review Letters 125, 047801 (2020), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/ PhysRevLett.125.047801

    2020

  9. [18]

    Pollard and G

    J. Pollard and G. P. Alexander, New Journal of Physics 23, 063006 (2021), ISSN 1367-2630, URL https://iopscience.iop.org/article/10.1088/ 1367-2630/abfdf4

    2021

  10. [19]

    L. C. B. Da Silva and E. Efrati, New Jour- nal of Physics 23, 063016 (2021), ISSN 1367-2630, URL https://iopscience.iop.org/article/10.1088/ 1367-2630/abfdf6

    2021

  11. [20]

    J. V. Selinger, Annual Review of Condensed Mat- ter Physics 13, 49 (2022), ISSN 1947-5454, 1947- 5462, URL https://www.annualreviews.org/doi/10. 1146/annurev-conmatphys-031620-105712

    2022

  12. [21]

    Subert, G

    R. Subert, G. Campos-Villalobos, and M. Dijk- stra, Nature Communications 15, 6780 (2024), ISSN 2041-1723, URL https://www.nature.com/articles/ s41467-024-50935-4

    2024

  13. [22]

    Sadoc, R

    J.-F. Sadoc, R. Mosseri, and J. V. Selinger, New Jour- nal of Physics 22, 093036 (2020), ISSN 1367-2630, URL https://iopscience.iop.org/article/10.1088/ 1367-2630/abaf6c

    2020

  14. [23]

    J. P. Sethna, D. C. Wright, and N. D. Mer- min, Physical Review Letters 51, 467 (1983), ISSN 0031-9007, URL https://link.aps.org/doi/10.1103/ 19 PhysRevLett.51.467

    1983

  15. [24]

    B. G.-g. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Physical Re- view Letters 110, 237801 (2013), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/ PhysRevLett.110.237801

    2013

  16. [25]

    J.-S. B. Tai and I. I. Smalyukh, Science 365, 1449 (2019), ISSN 0036-8075, 1095-9203, URL https://www. science.org/doi/10.1126/science.aay1638

    2019 doi

  17. [26]

    Wu and I

    J.-S. Wu and I. I. Smalyukh, Liquid Crystals Reviews 10, 34 (2022), ISSN 2168-0396, 2168- 0418, URL https://www.tandfonline.com/doi/full/ 10.1080/21680396.2022.2040058

    2022

  18. [27]

    Seˇ c, T

    D. Seˇ c, T. Porenta, M. Ravnik, and S. ˇZumer, Soft Matter 8, 11982 (2012), ISSN 1744-6848, URL https://pubs.rsc.org/en/content/articlelanding/ 2012/sm/c2sm27048j

    2012

  19. [28]

    Darmon, M

    A. Darmon, M. Benzaquen, S. ˇCopar, O. Dauchot, and T. Lopez-Leon, Soft Matter 12, 9280 (2016), ISSN 1744- 683X, 1744-6848, URL http://xlink.rsc.org/?DOI= C6SM01748G

    2016

  20. [29]

    Darmon, M

    A. Darmon, M. Benzaquen, D. Seˇ c, S. ˇCopar, O. Dau- chot, and T. Lopez-Leon, Proceedings of the National Academy of Sciences 113, 9469 (2016), ISSN 0027-8424, 1091-6490, URL https://pnas.org/doi/full/10.1073/ pnas.1525059113

    2016

  21. [30]

    Posnjak, S

    G. Posnjak, S. ˇCopar, and I. Muˇ seviˇ c, Scientific Reports 6, 26361 (2016), ISSN 2045-2322, URL https://www. nature.com/articles/srep26361

    2016

  22. [31]

    Posnjak, S

    G. Posnjak, S. ˇCopar, and I. Muˇ seviˇ c, Nature Commu- nications 8, 14594 (2017), ISSN 2041-1723, URL https: //www.nature.com/articles/ncomms14594

    2017

  23. [32]

    Posnjak, Topological formations in chiral nematic droplets (Springer Berlin Heidelberg, New York, NY, 2018), ISBN 9783319982601

    G. Posnjak, Topological formations in chiral nematic droplets (Springer Berlin Heidelberg, New York, NY, 2018), ISBN 9783319982601

    2018

  24. [33]

    Pollard, G

    J. Pollard, G. Posnjak, S. ˇCopar, I. Muˇ seviˇ c, and G. P. Alexander, Physical Review X 9, 021004 (2019), ISSN 2160-3308, URL https://link.aps.org/doi/10.1103/ PhysRevX.9.021004

    2019

  25. [34]

    Pollard and G

    J. Pollard and G. P. Alexander, New Journal of Physics 26, 063027 (2024), ISSN 1367-2630, URL https://iopscience.iop.org/article/10.1088/ 1367-2630/ad5759

    2024

  26. [35]

    R. D. Kamien, Journal de Physique II 6, 461 (1996), ISSN 1155-4312, 1286-4870, URL http://www. edpsciences.org/10.1051/jp2:1996192

    1996 doi

  27. [36]

    E. I. Kats and V. V. Lebedev, JETP Letters 100, 110 (2014), ISSN 0021-3640, 1090-6487, URL http://link. springer.com/10.1134/S0021364014140070

    2014 doi

  28. [37]

    Pajak, L

    G. Pajak, L. Longa, and A. Chrzanowska, Proceed- ings of the National Academy of Sciences 115 (2018), ISSN 0027-8424, 1091-6490, URL https://pnas.org/ doi/full/10.1073/pnas.1721786115

    2018 doi

  29. [38]

    D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, et al., Proceedings of the National Academy of Sciences 110, 15931 (2013), ISSN 0027-8424, 1091-6490, URL https://pnas.org/doi/full/10.1073/ pnas.1314654110

    2013

  30. [39]

    Pollard and R

    J. Pollard and R. G. Morris, Morse Theory and Meron Mediated Interactions Between Disclination Lines in Nematics (2024), arXiv:2408.01032 [cond-mat, physics:math-ph], URL http://arxiv.org/abs/2408. 01032

    2024 arXiv

  31. [40]

    O. D. Lavrentovich and E. M. Terent’ev, Soviet Journal of Experimental and Theoretical Physics64, 1237 (1986), ISSN 1063-7761, aDS Bibcode: 1986JETP...64.1237L, URL https://ui.adsabs.harvard.edu/abs/1986JETP. ..64.1237L

    1986

  32. [41]

    Kleman, Journal de Physique 41, 737 (1980), ISSN 0302-0738, URL http://www.edpsciences.org/ 10.1051/jphys:01980004107073700

    M. Kleman, Journal de Physique 41, 737 (1980), ISSN 0302-0738, URL http://www.edpsciences.org/ 10.1051/jphys:01980004107073700

    1980 doi

  33. [42]

    Y. A. Nastishin, M. F. Achard, H. T. Nguyen, and M. Kleman, The European Physical Journal E 12, 581 (2003), ISSN 1292-895X, URL https://doi.org/10. 1140/epje/e2004-00030-7

    2003

  34. [43]

    Ciuchi, M

    F. Ciuchi, M. P. De Santo, S. Paparini, L. Spina, and E. G. Virga, Liquid Crystals pp. 1–13 (2024), ISSN 0267- 8292, 1366-5855, URL https://www.tandfonline.com/ doi/full/10.1080/02678292.2024.2313023

    2024

  35. [44]

    Eremin, H

    A. Eremin, H. N´ adasi, M. Kurochkina, O. Haba, K. Yo- netake, and H. Takezoe, Langmuir 34, 14519 (2018), ISSN 0743-7463, 1520-5827, URL https://pubs.acs. org/doi/10.1021/acs.langmuir.8b02823

    2018 doi

  36. [45]

    F. C. Frank and W. T. Read, Physical Review 79, 722 (1950), ISSN 0031-899X, URL https://link.aps.org/ doi/10.1103/PhysRev.79.722

    1950 doi

  37. [46]

    Bezi´ c and S

    J. Bezi´ c and S. ˇZumer, Liquid Crystals 11, 593 (1992), ISSN 0267-8292, 1366-5855, URL http://www.tandfonline.com/doi/abs/10.1080/ 02678299208029013

    1992

  38. [47]

    Eremin, Liquid Crystals Reviews 8, 29 (2020), ISSN 2168-0396, 2168-0418, URL https://www.tandfonline

    A. Eremin, Liquid Crystals Reviews 8, 29 (2020), ISSN 2168-0396, 2168-0418, URL https://www.tandfonline. com/doi/full/10.1080/21680396.2020.1867662

    2020

  39. [48]

    K. S. Krishnamurthy, D. S. Shankar Rao, M. B. Kanakala, and C. V. Yelamaggad, Physical Review E 103, 042701 (2021), ISSN 2470-0045, 2470-0053, URL https://link.aps.org/doi/10.1103/PhysRevE. 103.042701

    2021 doi

  40. [49]

    Bouligand and F

    Y. Bouligand and F. Livolant, Journal de Physique 45, 1899 (1984), ISSN 0302-0738, URL http://www.edpsciences.org/10.1051/jphys: 0198400450120189900

    1984 doi

  41. [50]

    P. J. Ackerman and I. I. Smalyukh, Physical Review X 7, 011006 (2017), ISSN 2160-3308, URL https://link. aps.org/doi/10.1103/PhysRevX.7.011006

    2017 doi

  42. [51]

    G. H. Derrick, Journal of Mathematical Physics 5, 1252 (1964), ISSN 0022-2488, 1089-7658, URL https: //pubs.aip.org/jmp/article/5/9/1252/231422/ Comments-on-Nonlinear-Wave-Equations-as-Models-for

    1964

  43. [52]

    Niv and E

    I. Niv and E. Efrati, Soft Matter 14, 424 (2018), ISSN 1744-6848, URL https://pubs.rsc.org/en/content/ articlelanding/2018/sm/c7sm01672g

    2018

  44. [53]

    R. D. Kamien and T. Machon, Proceedings of the Na- tional Academy of Sciences 117, 24102 (2020), ISSN 0027-8424, 1091-6490, URL https://pnas.org/doi/ full/10.1073/pnas.2014402117

    2020 doi

  45. [54]

    Chaturvedi and R

    N. Chaturvedi and R. D. Kamien, Proceedings of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences 476, 20190824 (2020), ISSN 1364-5021, 1471-2946, URL https://royalsocietypublishing. org/doi/10.1098/rspa.2019.0824

    2020

  46. [55]

    Pollard, Ph.D

    J. Pollard, Ph.D. thesis, University of Warwick, Coventry (2020)

    2020

  47. [56]

    M. P. Rosseto and J. V. Selinger, Physical Review E 105, 024708 (2022), ISSN 2470-0045, 2470-0053, URL https: //link.aps.org/doi/10.1103/PhysRevE.105.024708

    2022 doi

  48. [57]

    V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. 20 Vasil’ev, Singularity Theory I (Springer Berlin Heidel- berg, Berlin, Heidelberg, 1998), ISBN 9783540637110 9783642580093, URL http://link.springer.com/10. 1007/978-3-642-58009-3

    1998

  49. [58]

    Machon and G

    T. Machon and G. P. Alexander, Proceedings of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences 472, 20160265 (2016), ISSN 1364-5021, 1471-2946, URL https://royalsocietypublishing. org/doi/10.1098/rspa.2016.0265

    2016

  50. [59]

    ˇCopar and S

    S. ˇCopar and S. ˇZumer, Physical Review E 85, 031701 (2012), ISSN 1539-3755, 1550-2376, URL https://link. aps.org/doi/10.1103/PhysRevE.85.031701

    2012 doi

  51. [60]

    Machon and G

    T. Machon and G. P. Alexander, Physical Review X 6, 011033 (2016), ISSN 2160-3308, URL https://link. aps.org/doi/10.1103/PhysRevX.6.011033

    2016 doi

  52. [61]

    Efrati and W

    E. Efrati and W. T. M. Irvine, Physical Review X 4, 011003 (2014), ISSN 2160-3308, URL https://link. aps.org/doi/10.1103/PhysRevX.4.011003

    2014 doi

  53. [62]

    D. A. Beller, T. Machon, S. ˇCopar, D. M. Sussman, G. P. Alexander, R. D. Kamien, and R. A. Mosna, Physical Re- view X 4, 031050 (2014), ISSN 2160-3308, URL https: //link.aps.org/doi/10.1103/PhysRevX.4.031050

    2014 doi

  54. [63]

    J. V. Selinger, Liquid Crystals Reviews 6, 129 (2018), ISSN 2168-0396, 2168-0418, URL https://www.tandfonline.com/doi/full/10.1080/ 21680396.2019.1581103

    2018

  55. [64]

    C. D. Hansen and C. R. Johnson, eds., The visualization handbook (Elsevier-Butterworth Heinemann, Amsterdam ; Boston, 2005), ISBN 9780123875822

    2005

  56. [65]

    B. G.-g. Chen, PhD Thesis, University of Pennsylvania (2012)

    2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.