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The knots in your shoelaces are familiar, but can you imagine knots made from light, water, or from the structured fluids that make LCD screens shine?

They exist, and in a new study, researchers created particle-like so-called ¡°vortex knots¡± inside chiral nematic liquid crystals, a twisted fluid similar to those used in LCD screens. For the first time, these knots are stable and could be reversibly switched between different knotted forms, using electric pulses to fuse and split them.

¡°These particle-like topological objects in liquid crystals share the same kind of topology found in theoretical models of glueballs, experimentally-elusive theoretical subatomic particles in high-energy physics, in hopfions and heliknotons studied in light, magnetic materials, and in vortex knots found across many other systems,¡± explains , director of the at the University of Colorado Boulder and a professor in CU Boulder¡¯s Department of Physics.

Fusion and fission of such topological structures are believed to occur deep inside quantum and magnetic materials that underpin many modern technologies. However, those processes unfold at scales too tiny and too fast to observe directly. Here, vortex knots appear at a scale that can be seen and controlled in real time.

By making these topological transformations directly visible and controllable, the team provides a physical testbed for mathematical ideas that until now have mostly lived only on paper. The results open possible new routes toward knot-based electro-optic and photonic technologies.

Bringing knot diagrams to life

In the 1860s, Lord Kelvin proposed that atoms might be tiny knots of swirling vortices that are stable because of their topological geometry rather than their material composition. That idea seeded the modern field of knot topology in mathematics, but physical vortex atoms remained theoretical.

Actual knotted vortices are a relatively new phenomenon, first produced in 2013 by Dustin Kleckner and William Irvine using water flow past a knotted template. These structures quickly relax into simpler forms and disappear due to a process called reconnection, where the vortex lines break and rejoin in new ways, changing the topology and ultimately, untying the knot.

Without long-term stability, they cannot behave like Kelvin¡¯s persistent atom-like topological structures, and they have not offered use in technological applications.

Smalyukh¡¯s group has long worked to create stable particle-like knots and new forms of metamatter from them.

Vortex knot fusion, shown under polarized optical microscopy and in numerical simulation

Their approach uses chiral nematic liquid crystals, materials whose molecules twist into a helical pattern that naturally supports knotted fields. In this study, by tuning the helical pitch of the molecules between 5 and 10 micrometers and confining the liquid between transparent, electrode-coated glass plates, the team created a stable environment for vortex knots.

Knots are created by holographic laser tweezers that locally melt small regions of the liquid crystal. As those spots cool, the material relaxes into twisted, knotted three-dimensional particle-like configurations known as heliknotons.

Indium tin oxide electrodes allow researchers to apply sub-second voltage pulses that modulate molecular alignment. By raising or lowering the voltage, they could make individual knots expand, shrink, merge, or divide.

¡°We unexpectedly observed that knot fusion is reversible,¡± says Darian Hall, first author of the study, who completed this work as an undergraduate at CU Boulder and is now a PhD student at UC Berkeley.

While fused knots tend to remain together to minimize free energy, the team discovered that rapidly reducing the voltage causes the knots to separate again. Hall notes that although identifying the right conditions required extensive testing, the team ultimately found specific electrical pulse durations and voltage levels that reliably control fusion and fission.

Night inside the mathematician¡¯s notebook

The team found that this physical system truly followed the rules of knot theory. When two knots fused, vortex-line segments with opposite twists canceled out and formed new composite knots, which later split apart when the voltage was reversed, all occurring at subsecond timescales.

These behaviors matched mathematical operations such as band surgeries and connected sums, showing that the liquid crystal was performing the same topological manipulations that mathematicians once only described in diagrams.

Co-author , professor at the University of Illinois at Chicago and a leading knot theorist, said the liquid-crystal knots reminded him of museum objects suddenly coming to life, much like in the film ¡°Night at the Museum.¡±

¡°I have been particularly interested in the least number of reconnections needed to unknot a knotted vortex,¡± he added. As such, the team mapped out how different types of reconnection lead to distinct outcomes. In this uniquely stable chiral liquid-crystal system, some reconnections do not simplify knots but instead can build new, long-lived structures, from composite knots to multi-loop links.

¡°Knotted structures in liquid crystals have more stability than vortices in water and they provide a new environment to study reconnection,¡± says Kauffman.

To confirm that these transformations were truly topological, the team tracked a mathematical quantity called the Hopf index, the system¡¯s topological charge, and found that it remained constant even as the knots fused or split.

The experiments and simulations also revealed a deep connection between chirality, or handedness, and topology. Reversing the molecular chirality of the liquid crystal host from left-handed to right-handed reversed the handedness and charge of every vortex knot inside. This showed that molecular chirality not only stabilizes the knots but also directly determines their handedness.

When math becomes matter

This work shows how, under the right conditions, particle-like knots could in principle be found anywhere, including in the pixels of the liquid crystal display that you might be using to read this article. This study¡¯s liquid crystal knots are an example of organized matter whose stability and behavior come not only from what it is made of but from how it is arranged.

Such topological organization is a foundational principle behind a paradigm spearheaded by Smalyukh called ¡°knotted chiral meta matter,¡± an approach to study and design materials whose function emerges from their knotted and chiral structure. These ideas inspired the creation of WPI-SKCM?, the , of which Kauffman is also an affiliate member. Members at the institute are working to understand knot topology and chirality across scales and disciplines and to harness these principles to build new forms of matter with pre-engineered properties.

In this case, liquid crystal vortex knots may offer new ways to control light such as imprinting it with robust topological structure, enabling properties such as orbital angular momentum transfer, information encoding, or new types of light-matter interactions.

Smalyukh emphasizes that ¡°the current trillion-dollar-per-year liquid-crystal-based industries globally are already well equipped to build their next-generation technologies based on switching and fusing knots instead of just smoothly rotating liquid crystal molecules.¡± He envisions that these knotted liquid crystals could open the door to radically new technologies, from unconventional computation, data storage, and advanced displays to telecommunications, microactuators and artificial muscles, all enabled by topological control from knot theory foundations.

Journal: Nature Physics
Title: Fusion and fission of particle-like chiral nematic vortex knots
Authors: Darian Hall, Jung-Shen Benny Tai, Louis H. Kauffman & Ivan I. Smalyukh
DOI: