For Elisabetta Matsumoto, knot theory is knitting theory.

Elisabetta Matsumoto holds the gear cube octahedron Jitterbug that she designed with mathematician Henry Segerman. Image source... Johnathon Kelso for The New York Times

Boston-On the eve of the March annual meeting of the American Physical Society, a "stitch'n bitch" meeting was held on Sunday during happy hour in the lobby bar of the Westin Waterfront Hotel in Boston.

Earlier in the day, North Carolina State University physicist Karen Daniels tweeted the notice of the gathering: "Are you a physicist engaged in knitting, crocheting or other fiber arts?" She asked. "I will be the one who weaves the torus." (The torus is a mathematical doughnut; she was inspired by a character in a friend's scientific paper.)

In the bar, with balls of yarn piled on the table, Dr. Daniels listened to design suggestions from a group of professional weavers, including Elisabetta Matsumoto, an applied mathematician and physicist from Georgia Institute of Technology, who was also the co-host of the gathering.

For Dr. Matsumoto, knitting is more than just a handicraft hobby that is good for health. She is embarking on a five-year project funded by the National Science Foundation, "What a tangled network we weave," to study the mathematics and mechanics of "the ancient technology of knitting."

Some of the oldest examples can be traced back to Egypt in the 11th century. However, despite several generations of practical and empirical knowledge, few studies on the physical and mathematical properties of knitted fabrics are in a way that can generate predictive models about the behavior of such fabrics.

Dr. Matsumoto believes that "knitting is coding" and yarn is a programmable material. The potential benefits of her research range from wearable electronics to tissue scaffolds.

During the happy hour party, she knitted a sample to demonstrate a plastic surgery technique called Z-plasty. This sample is used in her speech entitled "Twisted Topological Tangles" to be delivered at 8 AM on Wednesday morning. Although there were competing parallel conferences on "The Limit Mechanics of Balloons," dozens of physicists were still in attendance.

"I've been knitting since I was a kid," Dr. Matsumoto told her (mostly male) audience. "That's what I did to get along with my mother when I was a teenager. It's just a dream to turn all these things I learned and played when I was a kid into scientifically rigorous things."

As a first step, her team is enumerating all possible knitting stitches: "There will be a countable infinite number. We are now studying how to classify them."

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The investigation is based on the mathematical tradition of knot theory. A knot is a tangled circle-a circle embedded in an intersection that cannot be unraveled. (A circle without an intersection is an "open".)

"Knitting stitches are a series of slip knots, one after the other," Dr. Matsumoto said. The rows and columns of slip knots form a very regular lattice pattern, similar to crystal structure and crystal material.

Through the knot theory, Dr. Matsumoto is essentially developing a knitting theory: a cell stitch alphabet, a stitch combination vocabulary, and a grammar that controls the knitting geometry and topology—the elasticity of the fabric, or its "Emergency Resilience". "

When discussing the emerging characteristics of knitting, Dr. Matsumoto sometimes mentioned a butterfly, a vibrant blue morpho. Its color is the result of optical emergence, not the result of chemical pigments, but the result of structure. In fact, every wing is a metamaterial: it is covered with nano-scale layers and arranged in a pattern called a gyro surface. The wing absorbs most wavelengths of light but reflects blue.

Knitted fabric is also a kind of metamaterial. A piece of yarn has almost no elasticity, but when it is configured in a slipknot pattern-knitted and woolen patterns-there will be varying degrees of elasticity.

"Only based on these two stitches, these two basic units, we can make a complete set of fabrics, and each of these fabrics has a significantly different elasticity," Dr. Matsumoto told the audience.

She first combined her mathematics and furry way of thinking as a doctorate. Students, after enjoying a friend’s crocheting explanation of a hyperbolic plane (curly kale is an example of a vegetable) and wondering how to do it differently.

"It's not isotropic, which annoys me," she recalled the day before the speech. She can see where the crochet starts, and a true hyperbolic plane should have no starting point and no direction.

She thought, "I can solve this problem."

She crocheted a network of heptagons resembling lace to produce a more uniform effect. Since then, Hyperbolic Plane has been her partner. In April, she tattooed a hyperbolic spiral on her left shoulder-a wonderful spiral spiral, a bit like a seashell.

In her speech, Dr. Matsumoto passed her hand-knitted samples: stockings (standard knitted sweater, quite elastic, used for T-shirts); garter belt (stretcher); ribbing (most elastic); and seeds (not So resilient, but one of her favorites).

A considerable part of her audience also showed off their hand-knitted products-sweaters, hats, comfortable water bottles, uncertain works. Dr. Matsumoto's most precious hand-woven work is her "Dragon of Happiness" shawl (from the weaver Sharon Winsauer, also known as Crazy Lace Lady).

After two months of knitting, Dr. Matsumoto found a stitch on the dragon's beard that she had never seen before.

"In the dragon pattern, there are all these crazy stitches," she said-the stitches not only occupy one cell on the pattern grid, but also extend to many cells, seeming to follow a horizontal arrangement rather than the usual The arrangement, the vertical direction.

Her team's knitting theory will combine these and other stitch patterns, as well as intentional stitch defects and constraints, such as how the yarn bends, twists, and compresses; how many layers, how thick, and how "fluffy" it is.

Matsumoto refers to the "halo area of ​​the yarn, in which the short-lived hairy fibers protrude," Dr. Matsumoto said, and it changes the way the two yarns interact, their friction and energy exchange. "I would love to use the word'floofy' as a technical term to write a paper."

Dr. Matsumoto's presentation opened a three-hour meeting entitled "Fabrics, Knits and Knots"-this is the first time this topic has been discussed at the annual meeting of the American Physical Society.

"Sabetta is extremely creative and she is doing really complex mathematical work," said Pedro Reis, the conference organizer and head of the Flexible Structure Laboratory at the Swiss Federal Institute of Technology in Lausanne. "She also attracted a lot of people into this field, otherwise they might not think about science at all."

In his introductory speech, Dr. Reis encountered the microphone cord entangled. "This is a good example of why we really care about this topic," he said.

Dr. Reis worked hard to solve the problems of long backhand shoelace knotting, climbing knots, basket weaving, surgical stitching, and how to teach the art of surgical knotting to robots. During the meeting, his laboratory partners described how they used CT scans to detect the internal structure of the knot filaments and the friction generated when the filaments contact. After the meeting, Dr. Matsumoto took some of her swatches and sent him home.

Derek Moulton of Oxford University mentioned sailor knots, variants of DNA and protein knots, and worms that tie themselves to minimize dehydration. He continued to discuss "whether it is possible to physically realize a knotted filament with zero-point self-contact." In other words, can there be a knot without cross-point contact? (It can; try it with a piece of paper or a rope at home.)

Harvard applied physicist Thomas Plumb-Reyes showed his research on "brushing hair" to a standing audience.

"What's the matter with tangled hair?" he asked. "What is the best combing strategy?"

Shashank Markande, a student who worked with Dr. Matsumoto, reported on their stitch classification work so far. Together, they came up with a conjecture: all stitches that can be knitted must be ribbon knots. (The ribbon knot is a very technical entanglement.) They thought about an inference: Can all ribbon knots be woven?

As early as February, Mr. Markande (who only recently started weaving for the sake of science) thought he had found an example of a non-woven ribbon knot, using a knot and link software program called SnapPy. He sent a text message with a sketch to Dr. Matsumoto: "Tell me can weave this?"

Dr. Matsumoto was about to go for a run. When she came back, she was manipulating yarn everywhere, and she had already figured out an answer. "I think we can knit," she texted back. When Mr. Markand asked her how to do it, she added: “According to our regulations, it can be knitted, but knitting with needles is not a trivial matter.”

Mr. Markand later said, "I was very surprised. With my limited knowledge, I thought it couldn't be woven. But Sabetta managed to weave it."

For the Tangled Web project, most of the experimental knitted fabrics were produced by replicas of old-fashioned knitting machines from the 1970s, namely Taitexma industrial and household knitting machine model TH-860, operated by doctoral student Krishma Singal. This machine can also be programmed via punch cards—just like the jacquard weaving machine invented by Joseph Marie Jacquard in 1804, sometimes referred to as the first digital technology.

Dr. Matsumoto's team likes to think about how stitch patterns provide codes — codes that are more complex than binary ones and zeros — to create programs for the elasticity and geometry of knitted fabrics. The buzzword is "topologically programmable materials," said postdoctoral fellow Michael Dimitriyev.

He is performing computer simulations on knitted fabrics, inputting yarn characteristics and stitch topology, and outputting the geometry and elasticity of the actual finished product. "I am a killer for flexibility," he likes to say.

The first paper the team is currently working on will verify Dr. Dimitriyev’s simulation and Ms. Singal’s hard copy samples. Once the computer simulation is perfected, Dr. Matsumoto and her collaborators can extract the knitwear behavior equations and algorithms, and then put these equations and algorithms into the physics engine of computer game graphics or movies.

Pixar's "Brave" and "Monster Company" showed cutting-edge animations of hair and fur, but the yarn has not yet appeared in the spotlight. Fabric animation is still very trial-and-error and requires time-intensive supercomputers to render.

"This may go in that direction," Dr. Matsumoto said. This is a good yarn, although only at the beginning, the edges are still a bit fluffy.