Framed C-lines

Knots ubiquitously describe how looped threads are arranged in space. For this reason, when analyzed within a physical framework, knots are typically found within fields defined by regions that unambiguously form curves in three-dimensional space. These knotted curves have been demonstrated in systems such as the vortices of fluids³², the intensity nulls of scalar optical fields^12,13,14, and within the C-lines of optical polarization fields²¹. C-lines specifically consist of curves of pure circular polarization in monochromatic electromagnetic fields³³. One of their most distinguishing features relates to the structure of the polarization field in their close proximity. Namely, they are enclosed by polarization ellipses with a major axis that rotates by integer multiples of π along a closed contour surrounding the C-line. This trait is in display in Fig. 1a, b. For the case of paraxial optical beams, polarization is confined within the plane transverse to the beam’s propagation, e.g., the xy plane. As shown in Fig. 1a, this restriction constrains the plane over which this polarization axis rotation can be traced. Non-paraxial beams, however, can feature polarization vectors whose normal is not perpendicular to the beam’s propagation. As displayed in Fig. 1b, this normal vector in turn dictates the plane in which the axis of the ellipse completes a half rotation around the C-line. The presence of these rotations consists of the key structural feature considered while defining the framed knots reported in this work.

Depiction of the polarization field in the proximity of a C-line when the normal of the polarization ellipse is a and is not b parallel to the beam’s direction of propagation. Polarization ellipses with an axis perpendicular to the C-lines are displayed in bold colors and determine the orientation of the line’s framing when it forms a closed loop. c Vector components of a framed optical knot, which include a circular component with knotted intensity nulls, \({E}_{-}^{\,{k}\,}\), accompanied by a longitudinal field, \({E}_{z}^{\,{k}\,}\), with nulls determined by the topology of \({E}_{-}^{\,{k}\,}\). These two components form a nonuniform polarization field E^k which can be shaped into a framed knotted C-line by means of a perturbing plane wave \({E}_{+}^{\,{p}\,}\). d Trajectory of the resulting knotted C-lines (red) overlaid onto the trajectories formed by the intensity nulls of \({E}_{-}^{\,{k}\,}\) (pink) and \({E}_{z}^{\,{k}\,}\) (orange) for various plane wave amplitudes. e Framed knot structures arising from the superpositions shown in c where the knotted C-line is shown in red and its frame is shown in cyan.

A framed knot in three-dimensional space is a knot, i.e., a looped curve, equipped with a vector field called a framing. The framing is nowhere tangent to the knot and is characterized by a number, the framing integer, which is the linking number of the image of the ribbon with the knot. In other words, it counts the number of times the vector field twists (2π rotations) around the knot. Knotted ribbons generalize framed knots to an odd number of half-twists, e.g., knotted Möbius bands. Given the above definition, we define the framing of a closed C-line by the axis of the adjacent polarization ellipse whose axis is perpendicular to the C-line’s tangent. This concept is illustrated in Fig. 1a, b, where we embolden the color of the polarization ellipse surrounding the C-line whose axis is perpendicular to its tangent, thereby defining its framing. In the rare case where all axes are perpendicular at a certain point of the C-line, the polarization vector defining the framing can be interpreted as the one enforcing its continuity with the least amount of twisting. This concept in turn defines the framing attributed to a knotted C-line. As illustrated in Fig. 1c, the latter may be constructed from a knotted field, E^k, defined by a circularly polarized component, \({E}_{-}^{\,{k}\,}\), with knotted phase singularities, and a longitudinally polarized component, \({E}_{z}^{\,{k}\,}\), ensuring that E^k satisfies Maxwell’s equations³⁴. By superposing E^k with a plane wave with the opposite polarization helicity, \({E}_{+}^{\,{p}\,}\), knotted C-lines arising from the singular structure of E^k are created. As shown in Fig. 1d, e, increasing the amplitude of \({E}_{+}^{\,{p}\,}\) with respect to that of \({E}_{z}^{\,{k}\,}\) molds the resulting C-line into the knot formed by the phase singularities of \({E}_{-}^{\,{k}\,}\). Further discussions involving the dynamics of this process are provided in Supplementary Note 1. Note that \({E}_{z}^{\,{k}\,}\) is negligible for paraxial beams, which are the main experimental focus of this work. Hence, for such beams, the C-line aligns with the aforementioned knotted vortices regardless of the amplitude of \({E}_{+}^{\,{p}\,}\)²¹.

Braid representation

In addition to their well-discernible three-dimensional structures, knots can also be represented by mathematical objects called braids. Geometrically, braids consist of intertwined arrangements of strands that do not turn back on ground that is already covered. Due to Alexander’s theorem, every knot can be expressed as a closed braid. For instance, the trefoil knot shown in Fig. 2a can be expressed as the closure of the braid shown in Fig. 2b. The concept illustrated in these diagrams can be further extended to knots and braids formed in three-dimensional space. For example, the trefoil knot embedded in the torus shown in Fig. 2c can be obtained through a stereographic projection of the braid enclosed in the cylinder shown in Fig. 2d^14,35. One way to perform this projection is to express this braid as the zeros of a complex field. This field is explicitly written as a function of the complex coordinates (u, v), which relate to the spatial coordinates, (x, y, h), in which the braid is embedded through u = x + iy and \(v=\exp (ih)\). This braided field can in turn be transformed into its corresponding knot with a stereographic projection defined by

where (ρ, φ, z) are the cylindrical coordinates of the three-dimensional space in which the knot is now embedded. In essence, this projection wraps the braid defined over (x, y, h) into a knot in (ρ, φ, z) by connecting its two ends, thereby effectively mapping the h coordinate to φ¹⁴. Further discussions on how the coordinates of each space map onto one another are provided in “Methods.”

The above projection is heavily relied on when constructing knotted optical fields. In particular, a scalar optical field can be constructed by first matching its field along the z = 0 plane to that of the complex knot resulting from the projection of a braid as prescribed by Eq. (1). When this optical field is paraxial, then its formulation at subsequent z planes can be obtained by means of paraxial propagation methods¹⁴. This method can then be further extended to describe paraxial-knotted C-lines²¹ and full vectorial solutions to the optical wave equation³⁴. For instance, the knotted field \({E}_{-}^{\,{k}\,}\) in Fig. 1c is fundamentally constructed based on the closure of a braid embedded within the zeros of a complex field³⁴.

Because of its wide usage in obtaining knots from braids, we have opted to use the projection defined in Eq. (1) to obtain structures with properties that can more easily be related to the braid representations of the optical-framed knots considered in this work. Namely, we consider the torus \({{\mathcal{T}}}_{2}\) obtained from the projection of the cylinder \({\mathcal{C}}\) enclosing the three-dimensional representation of the corresponding braid. Then, we scale the dimensions of our knots such that their structure fits within the proximity of \({{\mathcal{T}}}_{2}\). We later apply the coordinate transformation provided in “Methods” on those of a curve formed by a knotted C-line. This transformation effectively cuts the knot along a given azimuthal angle and unwraps it, thereby mapping the φ coordinate of the knot to the h coordinate of the space where the braid is defined. During this process, the orientation of the knot’s frame is assured to be locally preserved. To illustrate this procedure, we apply it on the framed optical trefoil knot shown in Fig. 2e. The resulting unwrapped structure is displayed in Fig. 2f. From this transformation, information such as the twisting angle in the knots’ braid representations can be extracted. Here, the twisting angle consists of the azimuthal orientation of the ribbon in the frame where the normal is aligned to the unwrapped knot’s tangent. For instance, the twisting angle in each strand of the unwrapped knot shown in Fig. 2f can be found in Fig. 2g.

Prime encoding scheme

Given the ability to extract the twisting angle of an optical-framed knot, we propose the following scheme exploiting these structures as information carriers. The use of this method relies on a pair of numbers (α, β) where α is a positive integer, and β is a number both related to α and to the topological structure of the framed knot. The latter is given by

where k refers to a strand in a braid representation of the considered framed knot. d_k is the number of half-twists along the kth strand exhibiting half-twists, i.e., d_k = −∞ for untwisted strands. p_k is a prime number assigned to the kth strand. Finally, M = ∑_kd_k consists of the total number of half-twists in the knot’s frame. Further discussions exploring how α and β relate to braiding and twisting in framed braids are provided in Supplementary Notes 2 and 3. With these variables, we define the natural number

whose prime factorization can be seen to be determined by the considered braid representation. Further details regarding this decomposition are provided in Supplementary Note 3.

The above representation of the framed knot and one of its braids may therefore be exploited for encoding and decoding topologically protected information as follows. Alice would like to send Bob a message which is here obtained as an output of a certain program running on some initial inputs, the set of numbers, d_k, k = 1, 2, …, n. Running the program with this set is expected to yield Alice’s message.

Alice conceives her program and its inputs as a framed braid. She identifies an operation with a sequence of crossings in the braid’s planar diagram while the initial inputs are taken as the number of half-twists per strand. Alice has her program completely specified by the n-strand framed braid representation of a knotted ribbon K_A. To maintain some degree of privacy, she would like to send Bob K_A rather than the original framed braid. As further discussed in Supplementary Note 3 and implied in Fig. 3, she takes note of the fact that K_A may be complicated such as to conceal the original framed braid.

A framed braid on the right encodes a message—the output of a certain program specified by the planar diagram of the braid. In particular, the braid representation can be linked, as discussed below, to the prime factorization of a large integer N. An operation in such a program is identified with a sequence of crossings. Its inputs are taken as the number of half-twists per strand. To maintain privacy, the closure of the braid, i.e., the framed knot/knotted ribbon (in the case of even/odd number of half-twists, respectively), is transmitted instead of the braid itself. This allows the sender to complicate the message, if desirable, by adding an arbitrary number of Reidemeister-II and -III moves. The unique framed braid representation may be recovered on the receiver’s end by transmitting two additional numbers, α and β, alongside with the knotted object. In this example, we chose for a simple elucidation α = 2 and the first three primes p₁ = 2, p₂ = 3, p₃ = 5, one per strand in the braid (to showcase the scheme in the richer case of three strands, we preferred here the figure-eight knot, rather than the double-strand trefoil and cinquefoil knots). The corresponding numbers of half-twists in our example are d₁ = 3, d₂ = 2, and d₃ = 1, giving a total of M = 6 half-twists in the resulting framed knot. This is the topological invariant to be transmitted. The number β is subsequently computed according to Eq. (2). Once received (on the left) the framed knot can be associated with the previously encoded integer N; the number of half-twists M and the pair α and β are substituted into Eq. (3) to yield N, which here equals 518, 400. The prime factorization of N results from the actual number of half-twists per strand in the braid representation, \(518,400={2}^{{2}^{3}}\cdot {3}^{{2}^{2}}\cdot {5}^{{2}^{1}}\).

She then proceeds by performing the following steps. She first chooses a positive integer α. She then determines the framed braid representation of K_A. Doing so involves allocating the number of half-twists in K_A to different strands of the braid, i.e., setting d_k such that M_A = ∑_kd_k. Following this step, she assigns prime numbers p_k to strands exhibiting half-twists. Finally, she determines the number β according to Eq. (2). Once this allocation is completed, Alice proceeds by sending Bob her knotted ribbon K_A and the pair of numbers (α, β) in real time. Upon receiving these, Bob computes N_α,β(M_A) whose prime factorization unfolds d_k. To prevent the latter from being retrieved as an unordered set of integers, Alice and Bob rely on a previously adopted convention clarifying how the extracted d_k is assigned to distinct strands of the encoded braid. Bob can now recover the framed braid that was originally considered by Alice. For illustrative purposes, we summarize this protocol in Fig. 3.

Experimental generation

Motivated by this encoding scheme, we proceed with its application to paraxial-knotted C-lines generated in the following experiments. Such structures can be created by means of the folded Sagnac interferometer used in ref. ²¹, which is shown in Fig. 4a for convenience. This apparatus separates a uniformly polarized light beam into two orthogonally polarized components, each of which modulated by a spatial light modulator (SLM). The latter displays holograms in which both the intensity and the phase of the target optical field is encrypted³⁶. One component is modulated to produce a beam featuring knotted optical vortices¹⁴, such as \({E}_{-}^{\,{k}\,}\) shown in Fig. 1c in the limit where non-paraxial effects are negligible. The other is modulated to form a large Gaussian beam that uniformly covers the entirety of the knotted component, thereby effectively taking the role of the plane wave \({E}_{+}^{\,{p}\,}\) in Fig. 1c. Upon exiting the interferometer, the two beams are coherently added, thereby converting the knotted phase vortices of \({E}_{-}^{\,{k}\,}\) into paraxial-knotted C-lines²¹. The knot and its frame can then be reconstructed with polarization tomography measurements³⁷ enabling one to obtain the field’s polarization profile.

We use the above apparatus to produce both framed trefoil and cinquefoil knots. The holograms displayed on the SLM for this purpose are displayed in Fig. 4b along with the amplitude and phase of the fields that they are designed to generate. The latter are given in Eqs. (4) and (5) for the cases of the trefoil and cinquefoil knots, respectively,

where ϱ is a scaled and dimensionless version of the cylindrical radial coordinate, φ is the azimuthal coordinate, and a, b, s are parameters that determine the shape of the knot. For the trefoil knot, we considered parameters of a = 1, b = 0.5, and s = 1.2, whereas for the cinquefoil knot, we used a = 0.5, b = 0.24, and s = 0.65. These fields are obtained based on stereographic projection methods explored in ref. ³⁵ and are further discussed in Supplementary Note 4. As discussed in the latter, the selected parameters enable the creation of shorter knots. Furthermore, as emphasized in Supplementary Note 5, the frame of these knots is less disrupted by noise in the position of the C-lines arising from experimental imperfections. The framed knots of these fields expected from theory are shown in Fig. 5a, whereas the knots generated in our experiments can be found in Fig. 5b. Aside from minor perturbations that arise where the C-lines are born and annihilated at the knot’s extremities, we observe that the knots’ frames are in fairly good agreement with what is expected from theory. The unwrapped form of our experimental knots based on Eq. (1) is shown in Fig. 5c. We plot the corresponding twisting angle of these unwrapped knots along with the one expected from theory in Fig. 5d, where we observe once more that both strands in the structure are endowed with the same number of half-twists.

At this point, it is worth accentuating that the quantity of interest in Fig. 5d consists of the total twisting angle in the unwrapped knot. It might be tempting to treat the latter as one of the knot’s braid representations. However, due to the knot’s unwrapping, the number of half-twists in each strand may not exactly amount to an integer. Both ends of the braid are mapped from an azimuthal cross-section of the measured knot. Therefore, if the orientation of the frame at this cross-section is not the same for all parts of the knot, then the twisting angle of the strands in the unwrapped knot will not strictly amount to integer multiples of π. However, the sum of the twisting angles in each strand will amount to such a multiple given that the knot is a closed structure. This physical trait, in conjunction with the aforementioned (α, β) pair, in turn allows us to formulate the properties of the braid under consideration, which, for our purposes, consists of a purely algebraic entity. By taking this consideration into account and following the scheme outlined in Fig. 3, the knotted structures illustrated in Fig. 5 along with a given choice of (α, β) can be used to encode a braid representation of these knots.