## Abstract

Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number . The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.

## 1. Introduction

The defects in condensed materials have a significant effect on their physical properties. While their existence may be caused by impurities and flawed preparation processes, controlled manipulation of defects can enable tuning of the physical characteristics of the material. The ability of a medium to host defects is a side effect of symmetry breaking on the microscopic scale. As such, the defects are intimately related to the topological properties of degenerate microscopic degrees of freedom. In materials without broken translational symmetry, such as uniaxial and biaxial nematics [1], ferromagnets [2] and superfluid helium [3,4], the defects can be roughly classified by their dimension into point defects, line defects and walls [5]. Their classification with the aid of homotopy groups is a well-established and efficient way of describing their interactions and the conservation rules they obey [6–9]. However, the basic topological analysis does not account for the fine geometric details, which are under control of the free energy. Specifically for nematic braids—networks of closed nematic disclination lines [10–12]—recent development has shown that additional geometric constraints allow for a finer classification of nematic braids and simplifies the understanding of their rewiring [13,14]. The constraint of a fixed disclination profile that holds true for nematic braids is not universal and does apply necessarily to some frustrated [15] or chiral systems [16,17]. The unconstrained cases, however, should still allow topological analysis through recognizable geometrical primitives—a quest that inspired the work presented in this paper.

In the following pages, we develop a formalism that observes the nematic disclinations in a coordinate frame, *locally aligned* to the disclination line tangent. This approach allows us to decouple the director behaviour from the global coordinates and use distinctive geometric features to evaluate topological properties of nematic loops without need of a reference frame. Our interpretation, which uses the algebraic formalism of quaternions known from homotopy theory, reproduces known results for loops with the restricted profile, but also extends naturally to unrestricted disclinations with general variable profiles. The geometric reasoning behind it translates the abstract topological formalism into a form suitable for interpretation of practical data acquired from experiments and numerical simulations.

## 2. Overview

Before introducing any new results, we shall briefly overview the existing theoretical background regarding nematic defect loops.

A nematic liquid crystal consists of elongated molecules that are locally orientationally aligned [1]. In a continuum approximation, the order is described by the director ** n**, a unit vector field, pointing in the direction of average molecular orientation. The head-to-tail symmetry of the molecules causes the sign of the director to be ambiguous.

Topologically, the director field is a map from the coordinate space to the real projective plane —the ground state manifold (GSM).^{1} The fundamental group distinguishes disclination lines with half-integer winding numbers from states that are topologically equivalent to the non-defect state, but makes no distinction between different topologically stable disclinations [9].

If a disclination line is closed (disclination loop), it can carry a topological point charge, measured by the second homotopy group. It may also be linked by another disclination, which is measured by the fundamental group of the field encircled by the disclination loop. Disclination loop can be wrapped in a torus and classified by two *π*_{1} winding numbers (small and large cycles of the torus) and the *π*_{2} topological charge [18–21]. These pieces of information are connected by the action of the first homotopy group *π*_{1} on the second homotopy group *π*_{2}, which causes ambiguity in the *π*_{2} topological charge [22]. The relevant information for a closed disclination loop is held by an integer topological invariant , which contains information about linking and topological charge parity [23]. The result, established by Jänich [23], shows that a set of *n* disclinations obeys a law of the form
2.1where *q* is the topological point charge of the set of disclination loops, Lk_{ij} are linking numbers between loops and *i* and *j* iterate over all the loops. Odd values of *ν* correspond to disclinations that are threaded by other disclinations an odd number of times, while even indices correspond to those that are threaded by an even number, which also applies to unthreaded loops. This theory describes all existing topological properties of disclination loops. However, the topological index *ν* is an abstract attribute assigned to each of the loops, so the application to the systems that are found in experiments, simulations and theoretical models requires a way to relate this index to observable attributes.

If instead of a three-dimensional space, the molecules of the nematic are restricted to move and orient in a two-dimensional plane, the system exhibits point defects, which can have any half-integer winding number, the lowest ones being and [9]. These two-dimensional point defects represent a restricted set of cross sections that the disclination lines in three dimensions can have. The topological equivalence of all disclination lines in the three-dimensional case is caused by the ability of the director to rotate through the third dimension. If we prescribe that the director is perpendicular to the disclination tangent, which is excellently obeyed in well-researched nematic colloids [12,24], closed disclination loops can be described as ribbons that follow the rotation of the profile. Ribbons can be assigned an invariant called the self-linking number Sl [25,26], which quantifies how many times the disclination profile rotates around the tangent when we traverse the loop [13]. In a braid of many disclination loops, each self-linking number is a *fraction* of the form Sl=*m*/3, where the numerator *m* is odd if the loop is threaded by an odd number of other disclinations, and even if the loop is linked by nothing or by an even number of disclinations. The fractional nature of the self-linking number is a consequence of the threefold symmetry of the disclination profile, which makes the director field continuous after rotations by a multiple of 2*π*/3. The self-linking numbers of a set of *n* disclination loops with a profile obey the rule
2.2which bears resemblance to Jänich's rule (2.1). The integers Lk_{ij} are the linking numbers between loops and Sl_{i} are self-linking numbers of individual loops. A comparison of equations (2.1) and (2.2) suggests a connection between the index *ν* and the self-linking number. In the following section, we formally derive this connection and extend it to disclination loops with more general profiles.

## 3. The profile circles and quaternions

All disclinations in a three-dimensional nematic have in common that the director on a test loop around the disclination traces an irreducible path in the GSM. As the GSM is a sphere with an additional condition of having the antipodal points identified, the loop is topologically equivalent to half of a great circle, which can be symmetrically extended to the full circle (figure 1). As the vicinity of the disclinations is subjected to elastic free-energy minimization, the deviations from the perfect half-circle are usually small, which makes the representation with a circle not only topologically allowed, but also with a reasonable quantitative approximation.

A closed disclination loop can be wrapped in a torus, *S*^{1}⊗*S*^{1}, generated by revolving a disclination-encircling loop parameterized by *u*, around the disclination loop parametrized by *v*. For each value of *v*, the parameter *u* traces a different great circle in the GSM (figure 1*a*). Each circle is fully specified by its normal and a reference director at a chosen parameter, *u*=0 (figure 1*b*,*c*). The sense of circulation around the circle is important and prescribes the sign of its normal. For example, the profile corresponds to the normal of the circle being anti-parallel to the disclination tangent in the real space, as the director rotates in the opposite sense compared with the circulation of the enclosing loop (hence the minus sign in ). The circle is invariant to rotations by *π* around its normal because each point in the real space maps to two antipodal points in the GSM.

Let the parametrization of the director field on the torus be ** n**(

*u*,

*v*). When

*v*changes, the entire circle rotates, so the director at all values of parameter

*u*transforms with the same rotation, parameterized by

*v*, 3.1The

**(**

*n**u*,0) is simply a reference profile at a freely chosen point on the loop. This sets the initial rotation to unity, R(0)=1.

The rotations vary continuously with loop parameter *v* and thus describe a path in the space of all rotations in three dimensions. We will use the quaternion representation of the SU(2) group.^{2} The unit quaternions and obey the conventional structure relations , , and . A rotation around an arbitrary axis ** a** by an angle

*ϕ*can be expressed as 3.2Rotation by 2

*π*around any axis yields R=−1, which is the irreducible rotational motion that ends up in the same state where it started. Note that if we took the SO(3) representation, these irreducible paths would be considered equal to the identity transformation, and part of the topological information would be lost.

The continuity condition for a closed loop is satisfied if the director is continuous for *all* *u*, that is, if the circle we obtain after completing the loop at (*v*=2*π*) matches the circle at the beginning of the loop (*v*=0) up to a *π* rotation around the circle's symmetry axis. Because the SU(2) group covers the space of rotations twice, the set of possible cumulative rotations expands to
3.3The allowed rotations form a cyclic group , enumerated by an integer *ν*, which corresponds to the index introduced by Jänich (2.1). To understand this correspondence, consider that R(2*π*) is simply the total rotation of the director at any fixed *u* on a circuit around the disclination. This circuit measures the fundamental group of the nematic encircled by the disclination and detects linking with another topologically stable disclination line, if the rotation angle is a half-integer multiple of 2*π* (). R=1 and R=−1 denote an unlinked loop with even and odd topological charges, respectively (this topic is discussed thoroughly in [21]).

Each rotation can be decomposed into the rotation R_{0}(*v*) of the local coordinate frame relative to the global coordinate frame, and the rotation R′(*v*) of the director relative to the local coordinate frame,
3.4We set the local coordinate frame to keep its *z*-axis aligned to the disclination loop tangent and vary continuously along the loop (e.g. the Frenet–Serret frame). We also ensure our framing is topologically equivalent to a framing of a planar circular loop without torsion, so the local coordinate frame simply rotates by 2*π* when we traverse the loop: R_{0}(2*π*)=−1. This extra rotational offset is present for all the loops, so we can factor it out. The multiplication of by −1 simply shifts *ν* by 2 so an alternative parameter *ν**=*ν*+2 mod 4 can be introduced, rewriting Jänich's law (2.1) as
3.5We can see that the number of loops *n* plays a role in the conservation law once we factor out the constant offset that gives an odd topological charge to disclination loops with *ν**=0. These trivial loops include the Saturn ring defect, loops created by opening point defects into rings, and generally all disclination loops with a constant non-rotating profile. All the significant information is now encoded in R′(*v*), which is measured in a convenient disclination-aligned local coordinate frame. Note that the sign in front of the linking numbers Lk_{ij} is irrelevant under modulo 2 conditions.

## 4. Discussion

With the framework of mapping between local profile rotations and Jänich's indices set up, the effect of each geometric operation on the disclination profile can be translated into the quaternion language and interpreted topologically. We will demonstrate interpretation of general disclination loops as a sequence of segments with and disclination profiles and brief transitions between them.

First, we reproduce the known result for the disclination profile (2.2). In the relative frame along the curve, the director of a disclination profile lies in the *xy*-plane and can only rotate around the *z*-axis, as it has to stay perpendicular to the disclination tangent. Rotation of the profile consists of both the passive rotation of the coordinate frame and the active rotation of the director. As the director lies in a plane, it can be specified by an angle *α*, so the profile rotation by *ψ* yields a *ϕ*=3/2*ψ* rotation of the director, and subsequently the entire circle (figure 2*a*,*b*),
4.1In a loop with a self-linking number Sl, the profile rotates by a total angle of *ψ*=2*π* Sl, and equation (3.2) gives the total local rotation
4.2Comparison with yields the index *ν**=3Sl. Note the use of the shifted index *ν**, as we are working in local coordinates. The factor of 3 elegantly removes the fractional quantization of the self-linking number that stems from the threefold symmetry of the profile. This result immediately transforms the expression (3.5) to (2.2) up to modulo 2 ambiguities, and explains the correlation between the linking of disclinations and the self-linking number, discussed in [13].

As a measure of total rotation of the director profile, the self-linking number can also be used to describe disclinations with a rotating profile. Setting in (4.1) yields *ϕ*=*ψ*/2 and *ν**=Sl, which generalizes the self-linking number and a corresponding conservation law to disclinations.

There is more to disclinations than just rotations of a fixed planar profile around the tangent. An obvious possibility is rotation of the director around a perpendicular axis by *ϕ*=*Pπ*, where *P*=±1 is the parity that distinguishes left-handed and right-handed rotation by *π*. The result of such a rotation is the transition between and defect profiles (figure 2*c*) [27]. The axis of rotation in equation (3.2) can point at any angle *θ* in the local *xy*-plane, , which leads to
4.3

The rotations can be composed sequentially in the order they are encountered with increasing parameter *v*. A real disclination loop can have a profile that rotates around a gradually varying axis that is neither parallel nor perpendicular to the disclination tangent. However, the principal reasoning in our derivation is topological, so the exact geometric realization of the rotations is not important. For example, the location of a transition between profiles can be pinpointed to the place where the circle's normal (in the GSM) is exactly perpendicular to the disclination's tangent (in real space). Note how this approach compares vectors from both the GSM (director space) and the real space, contrary to the traditional homotopy theory, which is unaware of the real-space geometry of the sample. Whether the transition between and disclination profiles happens along a very short segment of the disclination, or takes the entire disclination length to complete, has no effect on the calculated index, as the director field in the second scenario can be smoothly combed to the first one without changing the topology. Consequently, rotations around the *z*-axis (R′_{z}) and rotations by *π* around a perpendicular axis (R′_{⊥}) are enough to describe all director profile variations up to a small smooth deformation,
4.4and
4.5In rotations R′_{z}, the angle *ϕ* represents the local rotation angle of the director, which relates to the *physical* rotation angle *ψ* of the profile: for a profile and for a profile. The use of the physical angles *ψ* is convenient because it is easier to read from numerical or experimental data, especially when using an appropriate visualization method [24].

Because of the presence of the unit quaternion in the perpendicular rotation, the transitions do not commute with the *z*-axis rotations,
4.6Rotation of the profile on the one side of the transition is equivalent to an opposite rotation on the other side of the transition. The product of all local rotations must satisfy the continuity condition (3.3). As a perpendicular rotation introduces a factor , they must always appear in pairs—if the profile turns from to it must turn back into . The offset in the axis orientations adds an extra amount of torsion,
4.7If two sequential rotations about the same perpendicular axis are performed with the same parity (twice by *π* or twice by −*π*), they yield
4.8which effectively changes the index *ν** by 2 and switches the parity of the topological charge between even and odd. Opposite parities naturally cancel out.

As an example, consider a disclination loop with a reference profile of at *v*=0 that rotates by *ψ*_{1} around its tangent, flips to a profile around an axis at an angle *θ*_{1} and with parity *P*_{1}, then rotates by *ψ*_{2} and flips back to around an axis at *θ*_{2} with parity *P*_{2}. The total rotation of the profile along the entire loop becomes
4.9The exponent must be a multiple of *π*/2 to satisfy the director continuity, which puts a restriction on the angles *ψ*_{1,2} and *θ*_{1,2}, which is a generalization of the self-linking number quantization. When this condition is met, the expression above can be compared with in order to extract the index *ν**. Figure 3 presents an example with a particular choice of parameters *ψ*_{1}=4*π*, *ψ*_{2}=2*π*, *θ*_{1}=*θ*_{2} and *P*_{1}=*P*_{2}, which amounts to a total rotation . This signature indicates that the loop must be linked by another disclination.

Note that the self-linking rotations *ψ*_{1,2} of and parts have opposite-signed contributions to the index *ν**. This is because of the lack of commutativity of the profile transitions and profile rotations. Taking the profile as a reference as *v*=0 (figure 3) fixes the sign that then alternates for all further contributions owing to the commutation relation (4.6). Taking a different part of the loop as a reference cyclically permutes the order of the transformations. In contrast to the disclinations with constant profile, where the self-linking number was unambiguously defined, regardless of the choice of the reference profile, the non-abelian nature of the profile transitions causes ambiguity. For example, if a part serves as a reference, the entire exponent changes sign. This does not affect the R′=±1 rotations (*ν**∈{0,2}), but exchanges the signs of for linked disclinations. The same effect is observed if the parametrization of any of the disclination loops is reversed, which also reverses the linking numbers in equation (3.5). This demonstrates the reason for requirement of the classification with homotopy groups to have a fixed reference—a base point [9]. The choice of reference is very important once more than one loop is present in the system.

Most of the disclination loops can be analysed directly by the described formalism. Wherever the normal of the circle traced by the director on the unit sphere is not perpendicular to the disclination tangent, it can be simply regarded as being a or a profile, and the director can be unambiguously combed to the perpendicular plane. The decomposition thus takes advantage of easily recognizable physical features of the disclination profiles and can be counted by hand if provided with a properly visualized director field. In singular cases, such as a twist disclination line, which is exactly half-way between the and profiles, but can still have a profile that rotates around the disclination line tangent, appropriate rules can be quickly retrieved using a procedure similar to the one described above, but starting from a different reference profile.

## 5. Conclusion

The topological properties of nematics are an interesting topic, crucial for interpretation and design of experiments. Although the basic, strictly topological formalism has been known for a long time, easy application of the theory to real-world examples requires geometrization of the basic topological concepts. By considering local director behaviour, we can read the topological information from recognizable shapes of the director field. In cases when the free energy imposes restrictions to the disclination profile, we can make a fine-grained classification using not only homotopy theory, but also additional geometric information.

We have decomposed a general disclination loop into a sequence of local rotations that must add up continuously to a rotation that satisfies the director continuity in a closed disclination loop. Quaternions are an analytically efficient way of enumerating the local behaviour of the director profile and give a tangible geometric meaning to the indices *ν* from Jänich's classification. The use of SU(2) rotations on the entire profile at once instead of only considering the local behaviour of the director field further reinforces the close relationship between the uniaxial and biaxial nematics, as the unified handling of the entire cross section removes the differences between both phases.

As an extension of the theory of defects, the introduced formalism describes a wider range of disclination networks. Notable examples of systems that exhibit transitions between different director profiles are confined cholesteric phases [17], optically induced defects [28] and transient defects that are created by quenching from the isotropic phase [29]. Studies of random blue phases [30] and entangled structures in random or structured pores [15] could also benefit from improved classification of disclination profile variations.

The presented geometrization procedure is related to the decomposition of surface textures developed in [31]. The technique translates the abstract algebraic description of defects to intuitive counting of distinctive features of the director field, which makes it easier to understand experimentally and numerically acquired data.

## Acknowledgements

This research was funded by the Slovenian Research Agency under contracts nos P1-0099 and J1-2335, NAMASTE Center of Excellence, and HIERARCHY FP7 network 215851-2. S.Č. was supported, in part, by NSF grant no. DMR05-47230. Both authors also acknowledge partial support by NSF grant no. PHY11-25915 under the 2012 KITP miniprogram ‘Knotted Fields’.

## Footnotes

- Received March 29, 2013.
- Accepted May 10, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.