# Knot theory realizations in nematic colloids

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Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved December 29, 2014 (received for review September 5, 2014)

## Significance

Knot theory is a branch of topology that deals with study and classification of closed loops in 3D Euclidean space. Creation and control of knots in physical systems is the pinnacle of technical expertise, pushing forward state-of-the-art experimental approaches as well as theoretical understanding of topology in selected medium. We show how several abstract concepts manifest elegantly as observable and measurable features in nematic colloids with knotted disclination lines. Construction of medial graphs, surfaces, and Jones polynomials is showcased directly on experimental images, and adapted for the specific system of colloidal crystals in a twisted nematic cell. Discussing the correspondence between topological concepts and experimental observation is essential for building the bridge between mathematical and physical communities.

## Abstract

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

From the invention of ropes and textiles, up to the present day, knots have played a prominent role in everyday life, essential crafts, and artistic expression. Beyond the simple tying of strings, the intriguing irreducibility of knots has led to Kelvin’s vortex model of atoms, and, subsequently, a more systematic study of knots and links in the context of knot theory (1⇓–3). As a branch of topology, knot theory is a developing field, with many unresolved questions, including the ongoing search for an algorithm that will provide an exact identification of arbitrary knots.

As knots cannot be converted one into another without the crossing of the strands––a discrete singular event––knotting topologically stabilizes the structure. In physical fields, this coexistence of discrete and continuous phenomena leads to the stabilization of geometrically and topologically nontrivial high-energy excitations (4, 5). Examples of strand-like objects in physics that can be knotted include vortices in fluids (6⇓⇓–9), synthetic molecules (10, 11), DNA, polymer strands and proteins (12⇓–14), electromagnetic field lines (15, 16), zero-intensity loci in optical interference patterns (17), wave functions in topological insulators (18), cosmic strings (19), and defects in a broad selection of ordered media (20⇓⇓–23).

Nematic liquid crystals (NLC) are liquids with a local apolar orientational order of rod-like molecules. The director field, which describes the spatial variation of the local alignment axis, supports topological point and line defects, making it an interesting medium for the observation of topological phenomena (20, 24). Defect structures in NLC and their colloidal composites (25) have been extensively studied for their potential in self-assembly and light control (26), but also to further the theoretical understanding of topological phenomena in director fields (23, 27). Objects of interest include chiral solitons (28, 29), fields around knotted particles (30⇓⇓⇓⇓–35), and knotted defects in nematic colloids (36⇓⇓⇓⇓⇓–42). Each of these cases is unique, as the rules of knot theory interact with the rules and restrictions of each underlying material and confinement. The investigation of knotted fields is thus a specialized topic where certain theoretical aspects of knot theory emerge in a physical context.

In nematic colloids––dispersions of spherical particles confined in a twisted nematic (TN) cell––disclination lines entangle arrays of particles into “nematic braids,” which can be finely controlled by laser tweezers to form various linked and knotted structures (38, 39, 43). In this paper, we focus on the diverse realizations of knot theory in such nematic colloidal structures. We complement and extend the classification and analysis of knotted disclinations from refs. 32, 38, 40 with the direct application of graph and knot theory to polarized optical micrographs. We further analyze the nematic director with constructed Pontryagin–Thom surfaces and polynomial knot invariants, which enables a comprehensive topological characterization of the knotted nematic field based on experimental data and analytical tools. We use a λ-retardation plate to observe and distinguish differently twisted domains in the optical micrograph, which correspond to medial graphs of the represented knots and contribute to the Pontryagin–Thom construction of the nematic director. Finally, we explore the organization of the space of possible configurations on a selected rectangular particle array and discuss the observed hierarchy of entangled and knotted structures.

## Colloidal Arrays, Knot Projections, and Tangles

We observe the TN cell with dispersed microparticles under the polarizing optical microscope (POM), sandwiched between crossed polarizers. Due to the light scattering on the defect core, the disclinations (line defects) in the nematic order that form around the particles are clearly visible and trace out a knot diagram (1, 2), as demonstrated in Fig. 1. In the projection seen by the microscope, we observe only two types of crossings: at each particle, the top and bottom disclinations cross perpendicularly; in the space between four neighboring particles there is a point where the disclinations can also cross, but they can also bypass one another. The experimental image thus constitutes a knot diagram, which is all of the information needed to further analyze the topology of the knot formed by the disclination loops (Fig. 1 *D*–*F*).

The NLC between the particles can be manipulated with laser tweezers to switch between different entangled states. These so-called “tangles” are top projections of the three different configurations of a rewiring site where two disclination segments approach closely in a tetrahedrally shaped formation (40). The possible tangles are the crossing tangle and the smoothings and (Fig. 1*E*). These tangles exactly correspond to the tangles studied in knot theory and enter the knot invariants via Skein relations (1), which we will exploit later.

The relationship between the 3D nematic field and the projected knot diagrams ensures that in the TN cell, we observe exactly these three tangles at the positions between the particles in a grid. Due to the alignment of the TN director field at the bottom and top plate of the cell, there is only one type of crossing, which simplifies the calculations and restricts the set of possible knot diagrams on a selected grid size. Depending on how the tangles are configured, the disclinations can form different knots or links, which are limited in complexity only by the size of the colloidal aggregate (38, 39). The curve traced by the disclination can be extracted directly from the experimental image and either identified by hand, or simplified with specialized software (44) to detect which particular knot or link is formed (Fig. 1*F*). However, the geometric restrictions based on the physics of the containing TN cell lead to interesting properties that allow a more elegant investigation of the knotted structures.

## Twist Domains and Medial Graphs

An intriguing feature of entangled colloidal structures in a TN cell is the microscopic twist domains locked between the colloidal particles and the defects. Adjacent twist domains, separated by disclinations, have opposite handedness and their director field orientation in the cell midplane differs by *C* and 2*A*). The twist domains of opposite handedness are, in the absence of intrinsic chirality of the NLC, degenerate in free energy. When observing a large entangled colloidal crystal between crossed polarizers with an inserted λ-plate (45), the oppositely twisted domains appear as blue-green– and yellow-gray–colored regions (Fig. 2). The colors are caused by the difference in the angle between the average director orientation inside a particular domain and the orientation of the λ-plate. Minor color differences between the samples are caused by small variations in the cell thickness.

The twist-domain colors are related to graph theory. A duality between the knot diagrams and edge-signed planar graphs implies a knot projection can alternatively be represented as a graph. A knot projection delimits space into “faces” that can be alternately colored with two different colors. The vertices (nodes) of the graph represent the enclosed faces of one color, and the graph edges represent the crossings between the faces (1). Two-dimensional, entangled, colloidal structures in a NLC provide a natural framework for the construction of a planar graph from the projection of the entangled defect loops and the colored domains among them (Fig. 2). We assign a graph vertex to each same-colored region and connect adjacent regions with edges. Each graph edge is labeled as “+” or “−” depending on the type of the crossing it passes through Fig. 2*D*). The result is a signed planar graph, a medial graph of the knot diagram.

Every tangle rewiring is a local operation that does not change the global alternating color pattern, but only connects or disconnects the color domains. The edges that correspond to the disclination crossings seen at the particles all have one sign, whereas the edges corresponding to the tangles have the opposite (Fig. 2 *C* and *D*). This makes the graph construction easier: not only are the colors given by the experiment, but the edge signs can be immediately filled in without checking their orientations individually.

The medial graph is ideal for a straightforward calculation of the Göritz matrix and, subsequently, the knot signature and determinant, which for nematics also helps to determine the number of distinct director field textures (34). The resulting graph can be further simplified by the Reidemeister moves adapted for the graph representation (1). Thus, a simplified graph can be constructed and then used to identify the knot or link from a lookup table, or with a calculation of further invariants. The graph can be constructed even if the colloidal array is not rectangular, and works for any colloidal aggregate bound by disclinations (Fig. 2).

## Pontryagin–Thom Construction

The colored region extracted from the λ-plate micrographs can also be considered a top projection of a surface bounded by the disclinations (Fig. 3). At the crossings, this surface contains “twisted bands,” where the surface turns the other side toward the observer. An important topological invariant, the Euler characteristic, is obtained from the colored region as *b* is the number of rows in an *t* and *v* are the numbers of and tangles, respectively. The expression encodes the fact that the and tangles add additional bands to the surface. The surface is only orientable if the medial graph associated with the same colored region is bipartite––if it contains no odd cycles (for an odd cycle, see Fig. 3*B*). In the orientable case, the surface is a Seifert surface of the knot and can be used to extract topological properties via the Seifert matrix, such as the knot signature, Alexander polynomial, the upper bound of the knot’s genus, and the lower bound on the unknotting number (3).

Regardless of its orientability, the extracted surface can also be used to construct the Pontryagin–Thom (P-T) surface––a surface that compactly, but uniquely, describes the nematic texture up to a smooth topology-preserving transformation (29). It is defined as the surface where the director field lies in a plane perpendicular to the far-field orientation, and colored by the angle of the director in this plane (Fig. 3 *B* and *D*). In general, the definition of the far-field director is ambiguous in a TN cell, but for a *y* direction in Fig. 3), it is only a minor and topologically insignificant simplification.

The alternating nature of the director orientations of the twist domains between the colloidal inclusions implies that the P-T surface in the top projection coincides with the colored regions in the λ-plate micrographs as can be seen in Fig. 3 *A* and *B*. This fixes the shape of the surface up to a homotopy, but not the colors representing the precise director orientation. Without experimentally or numerically provided director field information, we predict the director based on the fact that around

The P-T surface is bounded by line and point defects in the system, and for the point defects, the number of turns of the “color wheel” measures the topological charge. The surface acts as a reference that connects point and line defects that together amount to a zero total topological charge. The defects and particles that are topologically bound together with the same P-T surface cannot be separated without remaining tethered or undergoing a discontinuous rearrangement. The “topological entanglement” defined in this way matches the intuitive notion of entanglement. The disclinations ensure that the entire cluster of particles is topologically irreducible, in contrast with the more common situation, where the point defects, particles, and disclinations are bound up into neutral pairs, which in turn interact weakly via residual elastic interactions (26, 46, 47).

This approximate construction retrieves a topologically exact P-T surface without using any director field information, simply by inferring the disclination orientation from the experimental image. The resulting surface can be used to further investigate the topological properties of the knot, and in particular, the director texture that lies in the knot complement (34). Such a classification is more precise, as two isotopic knots may have a topologically distinct director field around them.

## Polynomial Invariants

The examples above focused on individual structures and could be analyzed by hand. However, a robust and automated knot classification is needed to analyze the entire multitude of possible knots on a selected lattice. For that purpose, we use polynomial knot invariants, which are easily calculable from the tangle representation via the Skein relations (1, 2). A particularly suitable choice is the Kauffman bracket polynomial, because it does not require an oriented knot diagram (48). In this section, we shortly summarize the routine, adapted to the geometry of colloidal lattices, first developed for the classification of knots in ref. 38.

The computation of the Kaufmann bracket boils down to a recursion that relates a particular knot projection to modified projections with one crossing replaced by a smoothing. The recursion stops when all of the crossings are removed, leaving a set of unknotted and unlinked loops. A correction factor is required to obtain the Jones polynomial, which is a true topological invariant (48). A closed-form expression for the Jones polynomial sums over all simplified diagrams*u* and *v* are the number of and tangles that replaced all of the crossings and *n* is the number of components that build up each specific final configuration with no crossings (Fig. 4) (2). Sometimes the polynomial is expressed in terms of a different variable, *w*) is extracted from an oriented diagram of the original knot, as a sum of *D*).

To treat the nematic braids as knot projections, the particles are interpreted as tangles. On the aligned grid, the calculation can be easily automated, but the diagram can first be simplified in several ways. Firstly, the crossings that belong to the leftmost and rightmost column of particles can be removed (Fig. 4*B*). The remaining diagram can be further simplified by performing the grid-aligned counterparts of the Reidemeister moves of the first and second kinds, as demonstrated in Fig. 4 *C* and *D*. These moves can remove a great deal of unnecessary crossings in an *N* is the number of particles. This is of vital importance because the computational complexity of the Kauffman polynomial is exponential in the number of crossings. For a

After the Jones polynomial is obtained, we can find which knot or link a given nematic braid encodes. For knots of low enough complexity, a comparison of the result with databases of known Jones polynomials gives a probable answer, even though the Jones polynomial is not unique for each knot. The hypothesized knot has to be additionally verified using other methods, such as the medial graph analysis we mentioned above. For the knots that are found on colloidal arrays of sizes up to at least

The nematic braids are related to mirror knots and curves, with an additional constraint of allowing only one type of crossing (49). The above simplifications with Reidemester moves are the counterparts of simplifications for mirror curves, except for the Reidemester move III, which has no easy local realization if there is only one type of crossing. The rectangular layout of the knots resembles Celtic knotwork patterns. They are also examples of Legendrian knots, because the

## Hierarchy of Entangled Arrays

With the reliable classification algorithm described in the previous section, we can observe the structure of the configuration space of all of the knots on a selected lattice. Besides the knot type, the *q* of the system in the form*n* is the number of components, **2** by

The size of the particle array fixes the topological charge. In a diagram with *n* on its axes, the possible structures that can be generated by different combinations of tangles form a checkerboard grid with linked and unlinked structures in alternating grid cells (38). This approach imparts a partial order to the ensemble of

Any array can be constructed by incrementally adding single rows and columns, usually with the use of laser tweezers. Adding a column preserves the knot topology if the added tangles are all of the type (Fig. 5, *Bottom Right*). On the other hand, the addition of a row adds a single unknotted component if all of the new tangles are of the type (Fig. 5, *Top Left*). Changing one of the new tangles into a or tangle attaches the new loop to one of the existing loops, restoring the topology of the original structure. Thus, a larger array includes all of the knots found on smaller arrays (Fig. 5).

The persistence of old structures under the addition of rows or columns results in a hierarchical structure of lower-dimensional subdiagrams contained in larger diagrams. The addition of a row or a column in a topology-preserving way shifts the

Fig. 5 shows a hierarchical decomposition of a

## Conclusion

Among physical fields that potentially allow knotting, liquid crystals stand out as the most versatile, allowing the greatest variety and complexity, as well as a fine experimental manipulation. Among many NLC complexes where knots have been observed with POM, we selected the

Besides the knot-related topological invariants, the disclination loops have their own topological rules to obey. The self-linking number, and essentially the entire matrix of linking numbers for a multicomponent link (27), decorates the plain mathematical knots with an additional internal structure. We demonstrated that the self-linking number and the number of components act as good additional classifiers that allow organized cataloging of the knotted structures. We observe an ordering that weakly couples the self-linking number to the knot type, especially for the chiral knots. Although there is no strict law connecting the knot topology to the self-linking number, the hierarchy is nevertheless qualitatively observable. Additional statistical analysis could also yield a metric for the average distance between two knots in terms of the number of tetrahedral rewirings. The number of possible structures increases exponentially with the array size, and the

In conclusion, colloidal aggregates in a TN cell reveal how a physical frustration can cause the system to present itself in a form that is especially suited for a selected set of theoretical approaches. The application of topological methods to a physical system requires adaptations on both the theoretical and the experimental sides and provides a rigorous formalism for handling the more complex aspects of the studied material. Unveiling of interesting new features can be expected for many otherwise well-known materials that have yet to be studied with topologically guided algorithms. For abstract branches of knot theory, an experimental realization can be helpful for understanding and finding alternative methods and is not to be disregarded. The ability to observe the theory in action is a nice opportunity both for education purposes as well as clarifying scientific ideas and revealing a different perspective on existing problems.

## Materials and Methods

### Sample Preparation.

We used a dispersion of *N*-dimethyl-*n*-octadecyl-3-aminopropyl-trimethoxysilyl chloride (DMOAP, ABCR). The colloidal dispersion was confined to wedge-like TN cells, made of 0.7-mm-thick glass plates, coated with transparent indium–tin–oxide and rubbed polyimide alignment layers (PI-2555, Nissan Chemical), spaced by a

### Experimental Setup.

Silica microspheres and defect loops in the NLC were manipulated with laser tweezers setup built around an inverted optical microscope (Nikon, TE-2000) equipped with water immersion objective (Nikor, NIR Apo

## Acknowledgments

The authors acknowledge stimulating discussions with B. G. Chen, R. D. Kamien, R. B. Kusner, M. Ravnik, and R. Sazdanović. We thank the Kavli Institute for Theoretical Physics at University of California, Santa Barbara, for their hospitality during the 2012 “Knotted Fields” miniprogram and acknowledge partial support by National Science Foundation (NSF) Grant PHY11-25915. This work was supported by the Slovenian Research Agency under Contracts P1-0099, P1-0055, J1-3612, J1-6723, and Z1-6725, and in part by the Center of Excellence NAMASTE. S.Č. acknowledges support from NSF Materials Research Science and Engineering Centers Grant DMR11-20901.

## Footnotes

↵

^{1}S.Č. and U.T. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: uros.tkalec{at}ijs.si.

Author contributions: U.T. and I.M. designed research; S.Č. and U.T. performed research; S.Č., U.T., and S.Ž. analyzed data; and S.Č., U.T., I.M., and S.Ž. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

## References

- ↵.
- Adams CC

- ↵.
- Prasolov VV,
- Sossinsky AB

- ↵.
- Cromwell PR

- ↵
- ↵
- ↵
- ↵.
- Moffatt HK

- ↵.
- Scheeler MW,
- Kleckner D,
- Proment D,
- Kindlmann GL,
- Irvine WT

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Vilenkin A,
- Shellard EPS

- ↵
- ↵
- ↵.
- Nelson D

- ↵
- ↵.
- de Gennes P-G,
- Prost J

- ↵.
- Poulin P,
- Stark H,
- Lubensky TC,
- Weitz DA

- ↵.
- Muševič I,
- Škarabot M,
- Tkalec U,
- Ravnik M,
- Žumer S

- ↵
- ↵
- ↵
- ↵
- ↵.
- Liu Q,
- Senyuk B,
- Tasinkevych M,
- Smalyukh II

- ↵.
- Machon T,
- Alexander GP

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Tkalec U,
- Ravnik M,
- Čopar S,
- Žumer S,
- Muševič I

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Scharein RG

- ↵
- ↵
- ↵
- ↵.
- Kauffman LH

- ↵.
- Jablan SV,
- Radović L,
- Sazdanović R,
- Zeković A

- ↵.
- Geiges H

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