Untangling pressing global problems using knots

Pure mathematics isn’t renowned as a field that most directly addresses challenges in the real world. But a new research centre at Hiroshima University in Japan is on a mission to overturn that perception.

Specifically, researchers there intend to show the world that knot theory is relevant to a diverse range of applications, and could be a powerful tool for developing desperately needed sustainable solutions for global problems.

“In pure mathematics, ‘knots’ are purely mathematical entities, but they are powerful concepts for studying real things,” explains pure-mathematician Yuka Kotorii, deputy director of outreach and dissemination at the International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM²), part of the Japanese government’s World Premier International Research Center Initiative (WPI).

“We can use knot theory to analyse the shapes of many things in nature, especially string-like objects,” she adds.

“While our research is highly fundamental, it has the bigger picture in view,” says WPI-SKCM² director, Ivan Smalyukh. “We aspire to use fundamental knowledge to contribute to a sustainable future.”

Cropping up everywhere

A branch of topology, knot theory studies closed curves in three dimensions. Mathematical knots can represent the knots we encounter in everyday life. They crop up in many scientific disciplines, as Smalyukh discovered while conducting research in condensed-matter physics and the life sciences.

“While working with topological solitons and singular defects in liquid crystals, magnets and biomaterials, I discovered that some of the most fascinating structures were underpinned by knots from knot theory,” Smalyukh says. “Interestingly, the chirality of the host media was found to lead to the energetic stability of these topologically nontrivial structures, making them behave like particles, in many ways resembling the behaviour of atoms and subatomic elementary particles.”

Chemist and material scientist Hiroshi Sato, a principal investigator at WPI-SKCM² with a secondary affiliation at RIKEN, had a similar experience. He encountered knots through his work on materials known as metal–organic frameworks, porous materials that exhibit unprecedented structural and chemical tunability.

“To our surprise, we discovered that metal–organic frameworks with knotted structures exhibit some interesting mechanical properties that we hadn’t anticipated,” says Sato.

While knots appear a lot in nature, a key aspect of Smalyukh’s vision for WPI-SKCM² is to develop knots in fields as designable building blocks of artificial matter¹, thus introducing a new research paradigm of ‘knotted chiral meta matter’. The chiral nature of this novel matter is critical. “Chirality stabilizes knots as particle-like objects in physical and biosystems, and is key for our institute’s vision,” notes Smalyukh. In this process, researchers cross-pollinate mathematical knot theory and chirality knowledge across disciplines and scales.

“We can use knot theory to design new types of materials with properties that can help realize a sustainable future,” he says. “For example, materials that can help reduce energy demand, address challenges related to diseases, and store data.”

Practical knots

One example of how knots can have surprisingly practical applications is transparent aerogels that reduce the energy lost through windows². More transparent than conventional windows, these aerogels are more thermally insulating than still air. They consist of porous nanostructured metamaterials that contain about 99% air by volume.

Since building heating and cooling accounts for about 40% of the energy generated globally, and much is lost through windows, the aerogels could have significant energy savings.

Additional examples of knotted materials with exciting potential applications for addressing global warming are stimuli-responsive porous crystals.

“Nobody knew what types of porous crystals could be realized using knotted structures,” says Sato. “So purely out of curiosity, we asked: what would happen if we assembled knotted structures into crystalline materials?” To his surprise, they obtained soft crystals that could be highly deformed by the gentle application of pressure³.

Far more than scientific curiosities, these crystals could lead to materials that have highly beneficial properties, including thermal insulation and other properties. “We believe these results could lead to the creation of innovative porous materials that can adsorb and desorb gas molecules such as carbon dioxide simply by pinching and releasing them with our fingers,” says Sato.

Because knots and chirality occur in so many fields, their combination makes an ideal rallying point for interdisciplinary research. That’s the inspiration behind WPI-SKCM² — exploring areas as diverse as liquid crystals, colloids, biopolymers, gels, magnets and quantum systems, subatomic matter, planetary science and cosmology while introducing a new research paradigm of knotted chiral meta matter.

Cross-fertilization

A key emphasis of WPI-SKCM² is interdisciplinarity — pure mathematicians working with researchers from a wide array of fields, including physics, chemistry, biology, Earth and planetary sciences, materials science and engineering. The goal of such interdisciplinary work is to generate technological innovations while also providing novel perspectives that can contribute to breakthroughs in individual fields.

“It’s a huge challenge, almost like communicating with people from a different culture or language,” says Kotorii. “But I find the interactions invigorating. The key things are to respect the other person’s vantage point and to be prepared to listen to them. And also a willingness to set aside your own sense of what is common knowledge.”

Sato concurs. “Initially, it’s very difficult to understand each other’s perspectives but through ongoing discussions, we can uncover something totally unexpected and new,” he says. “It takes a bit of time, but it can be extremely fruitful, and so it’s vital to be patient in the first stages.”

Using the analogy of a chemical reaction, Sato says that the initial activation energy of the collaboration is high, but the energy released when the relationship gets underway is much higher. “And so we need a catalyst — that’s what WPI-SKCM² is seeking to be.”

One way to increase interdisciplinary interactions will be to have everyone working under the same roof — a building for the institute that will be completed by the end of 2025. In addition to the close to 30 principal investigators and co-principal investigators of WPI-SKCM², there is also cooperation between researchers from more than 50 institutes worldwide, including MIT in the United States, the Max Planck Institute in Germany, and Cambridge University in the United Kingdom.

Smalyukh sees WPI-SKCM² as special. “The institute is unique in that it’s connecting people with very different backgrounds from different disciplines, from top centres around the world,” says Smalyukh. “It’s very natural for us to think about the big challenges that humanity faces.”

And for Kotorii, it’s an opportunity to prove that pure mathematics can provide a key to addressing global problems. “WPI-SKCM² is an initiative in which pure mathematicians, who are not renowned for working on practical problems, can contribute to sustainability,” she says.