Knots and links, which are closed curves in three-dimensional space, were first proposed by Lord Kelvin in 1867 [1]. Since then, they have been observed and utilized in a wide array of physical systems, ranging from classical fluid dynamics [2–5] to liquid crystals [6–8], plasmas [9,10], and various quantum fields [11–14]. This brief historical overview highlights the significance of our research, which builds upon these foundational studies to advance the field of optical technologies.Table 1.

Coefficients cl,p of the Hopf Link Knot with One of the Lobes Being Rotated around the z-Axis

Despite being ubiquitous in nature, knots were studied for a long time for their purely mathematical interest. One of the essential questions in knot theory since the 19th century was how to distinguish two or more different knots, i.e., the knots that cannot be transformed into one another through a continuous transformation. This challenge, known as the knot equivalence problem, can be to some degree addressed by so-called knot invariants, or “fingerprints” of the knot. Several knot invariants have been defined, including the crossing number, tricolorability, p-colorability, and various polynomial-based invariants [15–18]. One commonly used invariant is the minimum crossing number, although it does not always define the knot uniquely. For example, there are two different knots with five crossings and three knots with a minimum of six crossings. In addition, it is not obvious how to determine when this minimum number is reached. Nevertheless, the minimum number of crossings has been used as a parameter for the classification of knots.

While knot classification, equivalency, and discovery of new types of knots have been the subject of intense mathematical research for over 150 years, realizing them in a physical world presents its unique challenges. In a majority of naturally existing physical systems, such as knotted bio-molecules, including proteins [19] and DNA [20], or nematic liquid crystals [21], the type of a knot and the way it forms is strongly dependent on the energy considerations. In contrast, knots formed by optical vortex lines are unique because they do not possess associated energy [22] and are characterized by lines of zero intensity, around which the phase circulates and increases by 2π [23–25]. Consequently, their excitation relies on finding the correct set of parameters corresponding to forming an isolated knot structure and minimizing its sensitivity to external perturbations such as misalignment, experimental imperfections, aberrations, or turbulence.

The isolated optical vortex knots have been realized in laboratory experiments using a superposition of Laguerre-Gaussian (LG) beams with different weights and radial and azimuthal indices [23,26,27]. Notably, the singularity lines, characterized by both zero intensity and indefinite phase, of these LG beams are susceptible to experimental noise [23,28–32]. Therefore, one of the important questions is whether the optical knot stability can simply be guaranteed by the topological stability of their mathematical counterparts or whether their shape needs to be optimized [33], and if so, how their shape can be modified as needed.

While several general approaches for modifying knot shapes have been proposed, such as parameter tuning in the braid representation of a knot [34,35] and methods aimed at enhancing the contrast between singularity lines, including iterative numerical optimization and controlling over-homogenization [23,36], these techniques have been limited to specific symmetric configurations. They do not address the more general case where systematic and on-demand modifications of optical knot shapes are required. This work addresses this problem by developing a mathematically rigorous method for controlling the topology, size, and spatial orientation of optical knots or their parts. The proposed approach universally applies to any knot created using braided zero lines.

We start with the representation of a knot first proposed by M. R. Dennis et al. [23]. In this approach, a periodic complex scalar field, with braided zero lines, is embedded in a cylinder of height of 2π, within the (x′,y′,h) coordinate system, as shown in Fig. 1(a). We employ Lissajous curves as strands to create knotted and linked structures. The relationship between the coordinates x′,y′ for each j-th Lissajous strand can be mathematically expressed as follows: {xj′(h)=rjxcos(sjxh+ϕjx),yj′(h)=rjysin(sjyh+ϕjy),(1)where rjx and rjy correspond to the radii of the j-th strand, sjx and sjy are the winding numbers corresponding to the number of cycles the strands make around the cylinder’s center, and ϕjx and ϕjy are the initial phases or rotational angles of the strands. Without the loss of generality, we assume rjx=rjy=1, sjx=sjy=sj, and ϕjx=ϕjy=ϕj. Introducing the notations u(x′,y′)=x′+iy′ and v(h)=eih, the strands can be described as roots of the following complex polynomial: q(x′,y′,h)=q(u,v)=∏j=1N(u−vsjeiϕj).(2)

Next, by applying the inverse stereographic projection, which is defined by the complex coordinates for the three-sphere as $$\begin{gathered} u(x′,y′)→u(x,y,z)=x2+y2+z2−1+2izx2+y2+z2+1, \\ v(h)→v(x,y,z)=2(x+iy)x2+y2+z2+1, \end{gathered}$$(3)the coordinate h can be smoothly transformed into the azimuthal angle of the torus. This transformation results in the Milnor map f(x,y,z)=q(u(x,y,z),v(x,y,z)). The numerator p(x,y,z) of f(x,y,z) is a Milnor polynomial representing the desired knotted and linked zero lines of the complex field in the (x,y,z) space. To ensure that these polynomials are solutions of the paraxial wave equation in the z=0 plane that represent optical beams with a finite amount of energy, f(x,y,z) is multiplied by a Gaussian envelope exp[−(x2+y2)/2w2]. Here, the beam waist parameter w influences the knot configuration and should be sufficiently large to avoid the intersection of the optical knot with outer singularities originating from z=±∞ [23]. An example of such a Milnor polynomial, specifically for the case of N=2, s1=s2=1.5, and ϕ1=0, ϕ2=π, corresponding to a trefoil knot, is shown in Fig. 1(b). Figure 1(c) shows a reshaped trefoil knot in the Milnor space with each lobe shifted along the z-axis, and the red-colored lobe rotated counterclockwise. The dotted lines trace the original knot. The intensity and phase distribution plots show discontinuities along the lobes that result from rotating the portion of the strand colored in red [Fig. 1(a)], and lead to the discontinuities in that strand at the ends of the rotated section at h=±2π/3. Similarly, such discontinuities can also occur in the parts of the knot’s structure corresponding to the ends of the strands at h=±π when the parameters sj, ϕj, and rj are chosen such that not all strands ending at h=π have corresponding strand ends with the same coordinates x′ and y′ at h=−π.

In previous studies, the only knot parameter used for knot shaping was the Gaussian beam waist parameter w. However, a new approach is required to manipulate the individual parts of the knot, such as in the example shown in Fig. 1(d). To address this issue, we propose a method aiming at the direct modification of the inverse stereographic projection in Eq. (3), enabling us to divide strands into arbitrary segments and independently manipulate the shape of each segment. This manipulation leads to specific Milnor polynomials [shown in Fig. 1(c)], resulting in the desired shape of the optical knot [Figs. 1(d) and 1(e)].

To demonstrate the proposed approach, we start with the simplest example of the Hopf link that can be described by Eq. (2) with N=2, s1=s2=1, and ϕ1=0, ϕ2=π: qHopf(u,v)=(u−v)(u+v).(4)

In the cylindrical coordinate system, the corresponding Milnor polynomial for the Hopf link at z=0 is given by pHopf(R,φ,z=0)=1−2R2−4R2exp2iϕ+R4. The field described by this polynomial with a Gaussian envelope is shown in Fig. 2(a).

To introduce additional asymmetry into our Hopf structure, we execute a counterclockwise rotation of one of the loops (shown in blue in Fig. 2) around the z-axis. This is achieved by rotating the stereographic projection given by Eq. (3) around the z-axis by angle α as follows: x→xcos(α)−ysin(α),y→xsin(α)+ycos(α),z→z.(5)

The key aspect of this procedure is applying the transformation exclusively to one of the braids, specifically the loop we aim to adjust, and in our case, it was the second braid (second factor) in Eq. (4). Consequently, such rotation of the Hopf link results in a modified Milnor polynomial: qHopf(u,v)=(u−v)(u+veiα).(6)

The modification of the stereographic projection, as performed above, results in the desired rotation accompanied by an additional shift of the loops along the z direction. The following correction can be applied to compensate for this shift: z→z+z0 for each lobe separately. As a result, the intended rotation of one of the loops is achieved, breaking the symmetry of the initial Hopf link. The complete derivation of this process is provided in Appendix A.

An example of such rotation with α=π/6 is illustrated in Fig. 2(b). We observe the counterclockwise rotation of the blue loop by π/6 compared to the original Hopf link, shown in Fig. 2(a). Importantly, such reshaping preserves all other features of the Hopf link, including the position of the red loop and the structure of the link along the z-axis, as shown in the last column of Figs. 2(a) and 2(b).

The described approach can further be extended to a variety of transformations. For example, both loops of the Hopf link can be rotated simultaneously. Let us consider an example of such a rotation where one loop is rotated clockwise by an angle θ1, and the other counterclockwise by an angle θ2 around the x-axis realized using the following transformation: x→x,y→ycos(θi)−zsin(θi),z→ysin(θi)+zcos(θi),i=1,2.(7)

The detailed derivation of this transformation is provided in Appendix B.

The transformation described by Eq. (7) results in a compression of the Hopf link along the propagation direction and significantly reduces its length. In the case of θ1=−θ2=π/5, the link is shortened almost four times, as shown in Fig. 2(c). Note that this transformation can be particularly advantageous for simplifying the experimental measurements of the knots, as it results in the reduction of the required scanning range. In the extreme case of close to π/4 rotation, the resulting Hopf link is almost entirely confined to a single plane. For the detailed derivations, please see Appendix C.

It is worth noting that extreme transformations—such as maximal rotations where one lobe overlaps with another one or strong compressions—may lead to situations where the knot topology is altered, or the optical vortices do not recombine as expected due to diffraction effects. Exploring these limiting cases requires a detailed analysis of both theoretical and experimental considerations, which is an interesting direction for future work.

The proposed approach is versatile enough to transform specific parts of various optical knots. Next, we apply it to one of the lobes of the trefoil knot. The important step involves using the transformation exclusively to the selected part of the trefoil knot, i.e., a particular lobe that we aim to rotate. To determine which part of the braid representation corresponds to each section of the trefoil, we refer to Fig. 1(a), where the corresponding parts are clearly visible. Figure 1 shows that we need to alter the stereographic projection in Eq. (3) for only one of the lobes. For demonstration, we choose the red lobe (although any other lobe or combination of them could be selected), defined by the first braid from −2π/3 to 2π/3. The angle of each trefoil lobe being equal to 4π/3 [2π/3−(−2π/3)=4π/3 for the red lobe] can be understood from the fact that the whole knot in the braid representation has a length of 2π [see Fig. 1(a)] and consists of two braids covering three different lobes. As a result, the total length of the singularity line composed of three lobes is 2π×2=4π, thus making the length of each lobe equal to 4π/3. After applying the rotation transformation given by Eq. (5) to the stereographic projection responsible for this segment of the chosen braid, we obtain the corresponding Milnor polynomial, presented in Appendix D. In this polynomial, the phase of the field exhibits two discontinuities, occurring at −2π/3 and 2π/3 between x and y axes [shown in Fig. 1(c)].

Although the corresponding field configuration does not satisfy the paraxial wave equation due to the discontinuities mentioned above in the Milnor space, it still provides a reliable approach to finding a physical solution by decomposing it into a basis that satisfies the paraxial equation. The most commonly used basis for optical knots is the LG modes basis, which we utilize in this paper as an example. To perform the LG decomposition of the Milnor polynomial obtained above, we numerically integrate this polynomial with a Gaussian envelope over a range of LG modes in z=0 plane with different azimuthal l and radial p indexes. The exact expressions for LG modes and the integral expression can be found in Appendix A. The ranges of l and p values are chosen such that the total power of the electric field of the combined LG modes, calculated as ∬|E(x,y,z=0)|2dxdy, reaches 99% of the total power of the Milnor polynomial. The threshold of 99% power was used because adding extra LG modes does not result in significant changes to the knot’s structure. Since every LG beam satisfies the paraxial equation, their sum, which leads to the desired optical knot, also satisfies it. As a result, rotating the lobe by π/6 eliminates all discontinuities.

Following the necessary modifications of the lobes along the z-direction, detailed in Appendix D, we realize an optical trefoil knot with one rotated lobe. A comparison of the original trefoil knot and the one with a rotated lobe is shown in Fig. 3. We observe that the 3D structure of the knot remains largely intact, except for the counterclockwise rotation of one lobe (colored red) by π/6 angle.

The final example, showcasing the practical application of our method, aims at simplifying the experimental measurements of optical knots. To achieve this, we need to increase the contrast between the zero intensity of the singularity line forming the knot and the intensity in their immediate surroundings. The enhanced contrast makes it easier to see and measure each singularity. The most challenging scenario occurs when two singularities are close to each other, creating a low-intensity spot between them that visually merges the singularities, thus significantly increasing the required sensitivity of the experimental setups. Therefore, we will focus on increasing the maximum field intensity that separates closely spaced singularity lines. By moving the singularities apart, we aim to increase both the distance between them and the intensity contrast around them [23,33]. This approach redistributes the intensity without affecting the outer rim, enhancing the central region’s contrast while maintaining sufficient energy for the knot’s proper formation, thereby improving the measurability of the central singularities.

Another essential aspect that can improve the ease of knot measurements is aligning the singularity lines with the propagation direction [23]. This alignment enhances the accuracy of measurements by reducing sensitivity to the camera position along the z-axis, as each XY cross-section now changes more smoothly with the z coordinate, thereby decreasing the required resolution along z. To achieve this alignment, trefoil lobes need to be rotated with respect to their original axes to reduce the tilt angle of the lobes to the z-axis, as described further.

To illustrate our approach, let us focus on manipulating the red-colored lobe in Fig. 3(a). Note that the same procedure can be applied to any lobe with only a change in the axis of rotation corresponding to the direction of each lobe (x-axis for the red lobe). For the blue lobe, the x-axis is substituted with a line at an angle of 2π/3 relative to the x-axis. Similarly, for the green lobe, the rotation axis forms an angle of 4π/3 with the x-axis. These specific axes of rotation are indicated as dashed lines in the third column of Fig. 3(a). In the first step, we separate the central singularity lines, labeled as 2, 4, and 6 in Fig. 3(a), to ensure equal distances between all singularities. For the red lobe, we adjust the positions of points 1 and 2, moving them to the right. The same procedure is repeated for the other two pairs of points, namely, 3 and 4, 5 and 6, along their corresponding rotation axes. The distance of these shifts is kept the same for all three lobes and is denoted as x0. To shift the 1-2 pair of points to the right, we update the transformation given by Eq. (3) for the red part of the braid representation in Fig. 1 since those singularity points belong to that lobe. This region is located between the polar angles of −2π/3 and 2π/3, as mentioned above. This shift along the x-axis can be described as follows: $$\begin{gathered} u(x′,y′)→u(x+x0,y,z)=(x+x0)2+y2+z2−1+2iz(x+x0)2+y2+z2+1, \\ v(h)→v(x+x0,y,z)=2(x+x0+iy)(x+x0)2+y2+z2+1. \end{gathered}$$(8)

After applying the same procedure to all three lobes, we optimize the distances between all the lines to be equal in the z=0 plane, aiming to minimize potential reconnections of the singularity lines and enhance the maximum field intensity between them.

Subsequently, as a second step to further enhance the experimental readability of the knot, we aligned the singularity lines with the propagation direction. This was achieved by rotating the red lobe along the line where y=z=0, employing a method similar to the one used for the Hopf link, as described in Eq. (7). Analogous rotations were also applied to the other two lobes, each along their respective rotation lines. We incrementally adjusted the rotation angles of the lobes to find the optimal alignment and settled on an angle of π/6 as a practical example where the alignment was satisfactory, and the trefoil knot was not excessively shortened along the z-axis. The final structure results from the superposition of six LG modes and is displayed in Fig. 4(a).

To evaluate the effectiveness of this procedure, we compared our modified knot with another version tailored explicitly for experimental purposes using a numerical iterative method, as detailed in Ref. [23]. Figure 4(b) shows that the procedure proposed in this study leads to the singularity lines that are not only more distant from each other but also straightened and better aligned with the propagation direction. Moreover, the intensity contrast between the singularities increased significantly, from 13% of the total intensity to 49%, which should dramatically simplify the experimental characterization of the knot.

Finally, we note that the knot in Fig. 4(b) comprises five LG modes, in contrast to the one in Fig. 4(a), which consists of six LG modes, making their comparison somewhat challenging. Therefore, by discarding the mode with the lowest amplitude and retaining only the five strongest ones in the case of Fig. 4(b), we demonstrate that our method still achieves superior contrast between singularities, even with fewer modes. While the contrast slightly decreases compared to the values in Fig. 4(a), it remains significantly higher than in Fig. 4(b) (see Appendix E).

Next, we experimentally measured the optical knots, designed using the proposed approach, using a Mach-Zehnder interferometer, shown in Fig. 5. A vertically polarized, continuous-wave laser tuned at 532 nm was used to illuminate a phase-only Hamamatsu X10468 liquid crystal on silicon (LCOS) spatial light modulator (SLM), where the hologram of each optical knot complex field was encoded using an inverse-sinc method [37]. After selecting the first diffraction order in the Fourier plane of the SLM using an iris diaphragm, the middle longitudinal plane of the encoded beam was projected onto the complementary metal oxide semiconductor (CMOS) camera. The phase distribution was then retrieved by measuring four different interferograms with a π/2 phase shift between the signal and the reference beams as [38] φ=arctan((I4−I2)/(I3−I1)),(9)where Ii is the i-th interferogram with a phase shift of 0,π/2,π,3π/2. Using a delay line, the CMOS camera was scanned along the direction of light propagation (or the longitudinal length of the optical knot). Once the system is aligned, we perform five consecutive measurements of the middle plane z=0 to ensure the complex field is consistent throughout the experiment, with a standard deviation of the measured normalized fields of 6% among them. Here, the standard deviation is defined as STD=(N−1)−1∑i=1N|Ai−μ|2, where N is the number of realizations, Ai is the complex field amplitude, and μ is the mean of A. The three-dimensional structure is recovered for each optical knot by capturing the complex field in about 45 planes along the longitudinal direction.

Figures 6(a) and 6(c) show the measured phase and intensity (inset) distributions, together with the top and side (inset) view of the recovered three-dimensional structure of the original and rotated Hopf links after the procedure described by Eq. (5), respectively. The measured rotation angle is 4.8π/6, which agrees with the intended rotation. Figures 6(b) and 6(d) show the experimentally retrieved phase and intensity (inset) distributions and the three-dimensional topological structure of the original and rotated trefoil knot, respectively. Here, the measured rotation angle is 5.2π/6. Additionally, we experimentally validated the optimized trefoil and compared it with the method previously presented by Dennis et al. [23]. The measured phase and intensity distributions and the three-dimensional trefoil knot obtained using our proposed optimization method are shown in Fig. 6(e). Figure 6(f) shows the same plots for the trefoil knot obtained using the optimization described in Ref. [23]. Figures 6(g) and 6(h) highlight the difference in the intensity contrast around the singularities at the y=0 cross-section of the intensity profile of both trefoil knots resulting from these two different optimization methods, respectively. Following our approach, the intensity around the singularity was enhanced up to 38% of the maximum intensity distribution, compared to 15% using the method described in Ref. [23]. The superior performance of our proposed approach can also be highlighted by comparing the straightness of the singularity lines shown in the inset in the three-dimensional structure plots (outer blue line, for instance).

We proposed a method allowing for versatile manipulation of the optical knot and link shapes using a rigorous mathematical procedure, allowing for on-demand changes of the shape, orientation, and relative size of the individual segments within the three-dimensional knots or links, as well as their rotation and shifts. We theoretically and experimentally demonstrated the effectiveness of the proposed approach by realizing the prescribed rotation of the individual lobes of the Hopf link and a trefoil knot, as well as their compression. These results may find applications in the field of three-dimensional optical trapping and manipulation or subwavelength microscopy. Finally, we applied the proposed approach to optimizing the knot’s shape for the ease of experimental measurements, the problem initially considered in Refs. [23,33]. We demonstrated that the proposed technique allows for enhanced contrast between singularities, increasing their separation and improving the alignment of the singularity lines along the propagation direction. Our method, based on stereographic projection reshaping and thoroughly validated laboratory experiments, is versatile and should be applicable across various applications ranging from nanophotonics and quantum optics to advanced spectroscopy and sensing.