For tie aficionados, knots are an art form—and a surprisingly difficult math problem.

Boris Mocka believes that, at one point, he had invented more necktie knots than anyone else on the planet—so many that he started to call himself a “tieknotologist.” Most people who wear ties are familiar with the four-in-hand knot, and perhaps the Windsor and the half-Windsor. But there are many others: the Plattsburgh, the Cavendish, the Hanover. In 1999, two physicists published a book titled “The 85 Ways to Tie a Tie.” Their tally, however, was far from comprehensive. Mocka alone has created more than fifteen hundred knots. The Gardenia looks like a flower; the Wicker and the Mockatonic look like origami. The Riddler looks like a question mark, and the Exousia requires more than one tie. “I’m very obsessed with being original,” Mocka told me.

Mocka is not a physicist, or a menswear designer, but a doorman in my apartment building. His path to sartorial invention is as winding as his silken creations. He was born in Martinique in 1969 and was raised in Paris. His father died when he was two, and his mother moved to New York without him. He was adopted by a man who painted in his free time, and who inspired Mocka to pick up drawing. By the time Mocka turned ten, people were telling him that his artistic skills were a gift from God. “We knew better, because it’s called practice,” he said.

At eleven, Mocka joined his birth mother in New York, where she worked for the French consulate, and he eventually enrolled in the United Nations International School. His relationship with his mother was fraught, and he occasionally ran away and slept on the streets. (They no longer speak, and his mother could not be reached for comment.) Eventually, he befriended a family member of the sculptor Claes Oldenburg, who helped him get into a leading art school, the Cooper Union. Before Mocka matriculated, however, he was drafted into the French military and shipped to Martinique. He head-butted a drill sergeant, and spent time in a penal colony in French Guiana. With only a month left in his service, he wanted to be discharged; his assigned psychiatrist told him that pleading insanity was the only option, so he did.

Mocka told me that when he returned to New York, he was homeless and suicidal for a time. “I was going back and forth between friends’ houses until I would outwear my welcome,” he said. “I’ve slept in trains, I’ve slept in staircases.” To earn some money, but also to prove to himself that he could beat anyone, he fought in underground, bare-knuckle boxing matches, using a French form of kickboxing that he’d learned in the military. (He says he never lost.) He taught fighting lessons in the East Village, delivered marijuana, and worked as a porter. Finally, he spent a decade at the 2nd Ave Deli, where he met his wife, and worked his way up from cleaner to manager, which allowed him to rent his own apartment.

Mocka invented his first tie knot in 2013, when he was forty-four. His wife was pregnant and he needed health insurance, so he got a union job as a doorman, at an apartment building where one of his friends worked. The night before his first day, he stayed up late learning to tie the Windsor knot, hoping to make a good impression.

When Mocka showed up for work, his friend was getting ready to go home. “I was overly excited,” Mocka told me. “I kept being, like, ‘Don’t you see how cool this is? This is the perfect Windsor.’ ”

His friend was unimpressed. “You know, you might want to look up the Eldredge knot,” the friend said. “If you go to YouTube, you’ll find videos.”

The Eldredge knot has the upside-down-triangle shape of a traditional knot. The surface of the triangle, however, is split into five overlapping sheaths, which make the knot look almost like a braid. While learning to tie it, Mocka made mistakes. “But those mistakes I liked better than the knot itself,” he said. He started making videos to keep track of the moves.

Soon, Mocka found an online community devoted to tie knots. He discovered knots like the Merovingian, which was named for a character in the 2003 film “The Matrix Reloaded”; it’s unusual because the tail is knotted in front of the blade, as though the tie is wearing a smaller tie. On YouTube, he started following tievangelists like Patrick Novotny, a Canadian realtor who wears uncommon tie knots to stand out while networking. Mocka ultimately created more than a thousand videos of his own. He established or articulated dozens of basic techniques: twisting, swirling, spiralling, tunnelling, snaking, flaring. He’d sometimes wake up in the middle of the night with a new idea, and rouse his sleeping wife by crying, “I got it!”

In the small world of tie-tying, Mocka became a minor celebrity. One prolific knot designer, a South African court interpreter named Mabu Letsoalo, called him a genius and a major influence on him. Novotny, whose tie-themed YouTube videos have more than twenty million views, told me, “I think everybody in the tie community could probably look toward Boris for inventing the most knots.” This is a surprisingly difficult thing to adjudicate, however. Mocka said that he reached out to Guinness World Records, but the organization said that a record would be too hard to establish; all of its records must be measurable, breakable, standardizable, and verifiable. Ties raise questions. Is any knot in a tie a tie knot, or are there rules? Can you turn an old knot into a new knot simply by adding a twist or a loop? When you combine two knots, do they become one? These are not just aesthetic questions—they’re math problems, too.

The La France

Knot theory falls within the mathematical field of topology, which explores fundamental properties of mathematical objects. Topologists treat shapes as though they are made out of rubber; they see cubes and spheres as the same, because one can morph into the other without any gluing or tearing. (Mathematicians sometimes joke that a topologist can’t tell the difference between a doughnut and a coffee mug: each has one hole.) According to topologists, a knot is just a closed loop. The simplest nontrivial knot is the trefoil, which is the same as an overhand knot, but with the string’s ends joined; on paper, it looks like someone drew a three-leaf clover.

For most of their history, tie knots and knot theory developed in parallel. The cravat, a type of scarf, gained popularity in Europe in the sixteen-hundreds, and an 1828 book, “The Art of Tying the Cravat,” included thirty-two cravat-tying lessons. The modern necktie emerged in the eighteen-fifties, with little function beyond fashion. The original knot, the four-in-hand, is said to have taken its name from a way of driving a horse carriage. In the nineteen-thirties, the larger Windsor knot became popular; it was named for the Duke of Windsor—although he actually wore the four-in-hand, just with large ties.

The word “topology” comes from the German mathematician Johann Listing, who wrote about knots in an 1847 book. Knot theory took another step forward in 1867, when the physicist Lord Kelvin—inspired by smoke rings blown by his colleague Peter Guthrie Tait—theorized that atoms were knotted vortices in the ether. Tait soon enumerated all the knots that had up to seven crossings. (There were fifteen.) Although they were proved wrong about atoms, the mathematics of knots became a specialty of its own.

Knots remain a knotty problem. Mathematicians long thought, for example, that there were a hundred and sixty-six knots with ten crossings, and they used tables depicting every one. Then, in 1974, a Harvard law student with an undergraduate math degree, Ken Perko, browsed one of these tables and noticed something funny: two of the knots were the same. (They are now known as the Perko pair.) Things get even more tangled—or maybe less—in higher dimensions. A knot in three dimensions comes undone in four; meanwhile, in four dimensions, you can knot spheres.

The art and science of tie knots finally started to converge in the nineties, when Thomas Fink and Yong Mao, physicists who were then working at Cambridge, learned about the invention of a tie knot by an American named Jerry Pratt. A clothier declared it “the first new knot for men in over 50 years.” Fink and Mao decided to do for tie knots what Tait had done for knots in general: catalogue them. “The discovery of a new tie knot, evidently, is a rare event,” they wrote in “The 85 Ways to Tie a Tie.” “Rather than wait another half-century for the next knot, we considered a more formal approach.”

The Ganesha

To apply knot theory to neckties, first imagine that the ends of the tie are fused after the knot has been tied, creating a closed loop. Then consider that a tie, unlike a traditional knot, must wrap around a person’s neck and drape against their chest. These two features—the neck and the drape—make tie knots a category of their own. According to standard topology, the four-in-hand and the Windsor are identical; they’re just the trefoil. The half-Windsor and the Pratt, meanwhile, are both unknots—simple rings. But, in practice, the four necktie knots are different.

Fink and Mao imagined a tie draped over a person’s shoulders, with the tail, or narrow end, on the left, and the blade, or wider end, on the right. They created a special notation for the ways one can move the blade, along with rules about sequence. They limited their tie knots to nine moves, ending with the blade coming up toward the chin and tucking down through a loop. Using their notation, what takes minutes to explain in a YouTube video can fit into a short string of characters. The four-in-hand, for instance, is L⊗R⊙L⊗C⊙T: left in, right out, left in, center out, through. They deemed thirteen of their knots to be aesthetically pleasing based on mathematical symmetry and balance. The four-in-hand, the Windsor, the half-Windsor, and the Pratt turned out to be among them; as far as they could tell, the other nine had never been described before.

Mikael Vejdemo-Johansson, a topologist now at the City University of New York, read Fink and Mao’s papers in 2013. As a mathematician who wore bow ties in his late twenties, and eventually graduated to neckties with fancy knots, he was their ideal audience. He found their research dissatisfying, however. Fink and Mao looked at knots with only flat surfaces, which excluded more complex creations like the Eldredge and the Merovingian. “Fink and Mao basically just define everything interesting out of the way,” Vejdemo-Johansson told me.

Vejdemo-Johansson knew that there is an infinite number of things you can do with a tie. Since most are too boring or weird to count as a tie knot, you have to make rules—but the rules also can’t be too strict. He decided to undertake his own study of ties with several collaborators. They decided that knots could have up to thirteen moves, could tuck through multiple loops, and did not need to end with the same sequence as Fink and Mao’s knots. They also decided that the tail, not the blade, could take the lead, which is the only way to tie an intricate knot like the Eldredge.

Mathematicians and computer scientists often study formal languages, which have their own symbols and rules about sequence. Vejdemo-Johansson described his tie knots as a distinct language, complete with its own notation and grammar. He and his colleagues showed that their grammar was inherently more complex than Fink and Mao’s, because some moves worked only in combination with other moves: you have to form loops before you can tuck a tie through them. But it wasn’t too complex for computational tools to enumerate all of the “sentences,” or knots, in the language.

In 2015, the team published a paper called “More Ties than We Thought.” Their final tally was a staggering 266,682. An accompanying Web site, tieknots.how, randomly generated knots and tutorials. Because the researchers did not apply any aesthetic rules, however, many of their knots were ugly. Math had yet to conquer art.

The Pectoral

For a while, Mocka made his ties into a small business. For fifty dollars, you could mail him a tie, and he’d tie a knot and mail it back. For two hundred dollars an hour, he’d tie one on you. This February, Mocka self-published a five-hundred-page book, “How to Tie ‘The’ Knot,” which explains dozens of techniques and gives instructions for sixteen signature knots; he sells the hardcover for two hundred dollars. But Mocka’s inventive period peaked around 2015. He needed to focus on supporting his family, and he was frustrated that tie fans were copying his work without proper credit. One designer, after hearing that Mocka wanted to set a world record for inventing the most knots, started competing with him, “making knots that make no sense,” Mocka told me. “They’re kind of messing up the game.” In the end, he made nearly all of his YouTube videos private.

In June, I invited Mocka and Vejdemo-Johansson to my apartment so they could meet for the first time. The mathematician arrived first, wearing a blue utility kilt, a formal vest and shirt, and a Trinity tie knot, a modern classic whose façade is divided into three symmetrical panels. He had a ponytail and a graying goatee.

At the time, Mocka was working as a porter in my building; because he was off duty, he arrived not in his usual work shirt, which was embroidered with the name “Boris,” but in a T-shirt and khakis. (Recently, he was promoted to doorman and began wearing fancy tie knots to work.) He had a shaved head, a graying beard, and no tie.

Sitting at my dining table, Vejdemo-Johansson flipped through Mocka’s book. He seemed excited: “I can already see things that are definitely not covered by the notation.” Apparently, there were even more ties than “More Ties than We Thought” had thought.

“I’m not a mathematician,” Mocka said. “I’m actually here to find out if I’m delusional.” He laughed, and then started to explain some of the techniques in his book.

Some of Mocka’s moves, Vejdemo-Johansson said, added geometric complexity without making a knot topologically different. Others added extra crossings. Either way, Vejdemo-Johansson went on, math hadn’t captured them yet. Mocka had effectively invented a new dialect, and its grammar was even more complex than the ones that professional mathematicians were using.

“You and I are operating in exactly opposite directions,” Vejdemo-Johansson told Mocka. “My goal is to pick a box and understand it. Your goal is to pick a box and transcend it.”

Mocka said that he wished he could teach a computer to create interesting new designs with his techniques. But Vejdemo-Johansson didn’t think a computer could pull it off. Computers are good at following rules; Mocka is good at breaking them. “You have a full world model,” Vejdemo-Johansson said. “You have understanding you’ve been honing for over fifty years, of how the world exists around you and how you interact with it.”

He cited Mocka’s Sparkling knot, which cradles a shiny object like a coin. An A.I. might not know the difference between originality and outlandishness; it might riff on the Sparkling knot by trying to insert the wheel of a car, or the tie-wearer’s wrist. Even with human feedback nudging it toward elegant knots, an algorithm might struggle to find one. In the space of all possible knots, Vejdemo-Johansson said, “my intuition is that interesting ties are too sparse.” The most elegant knots may be one move away from being ugly, or simply coming undone.

“The book was going to be called ‘A Million-Plus Ways to Tie Knots,’ ” Mocka said. “I didn’t have the scientific backup to confirm that.”

“It’s easy,” Vejdemo-Johansson replied. His paper described a quarter-million knots; the option of adding even a few of Mocka’s moves expands that number far beyond a million. “A one-line proof,” he said. The two men were now mathematical collaborators.

The Orb

I asked a few other mathematicians what they thought of Mocka’s knots. When Nicholas Scoville, a topologist at Ursinus College, in Pennsylvania, joined me on Zoom, he was wearing a tie that used one of Mocka’s techniques: folds created three little cups, as though the knot were meant to catch crumbs. Scoville later looked at Mocka’s book and considered what would happen if Mocka’s various folding moves were combined with Vejdemo-Johansson’s grammar. “I can’t imagine the number of new tie knots this would generate,” he said.

Elizabeth Denne, a mathematician at Washington and Lee University, told me that she likes knots because ordinary people can relate to them. “It’s easy to tell you what I’m doing,” she told me. “But the math that you need can be really full-on.” It’s possible to study knots using geometry, algebra, combinatorics, and analysis. When Denne and two students published a paper about Fink and Mao’s eighty-five knots, in 2021, they proved several things about them—for example, that they’re prime knots. (This means that if you slice them in two, you’ll be left with one trivial knot and one nontrivial knot.)

During a video call, Denne picked a knot from Mocka’s book. “Boris’s work is, like, a whole other kettle of fish,” she told me. “I don’t think this one’s prime, because he’s got, like, junk going on.” She pointed to one part of the knot in particular. “He’s doing half hitches over here. And then he’s threading that through. This thing that he’s done on the side—that’s a separate knot, right there. So this is not a prime tie knot.” At least, she didn’t think so.

Many more math techniques would be needed to describe Mocka’s art. Knot theory often treats strings as one-dimensional—but ties have width, too, making them more like ribbons than threads. For this reason, a twist in a tie could have mathematical significance, just as it does for a famous topological object called a Möbius strip. Denne specializes in physical knot theory, which considers the real-life width or volume of things like ribbons or ropes—but doing so “is really, really hard,” she told me. “It’s almost like your everyday knots are too hard.” No one has managed to add friction to the equation. The Merovingian will have to await further study.

I wondered whether the mathematics of tie knots could be applied anywhere else—to other areas of science, or to the real world. “I don’t know what the application is beyond tie knots,” Denne said. “But, then, nobody thought about origami being useful.” She cited NASA technologies that were inspired by folded paper, such as a star shade that unfolds in space. Contemporary physicists apply knot theory to quantum mechanics and the winding paths that particles take through space-time. Knots have been observed in turbulent fluids and electromagnetic fields. Chemists have imagined molecules tied into knots to produce specific properties. And biophysicists sometimes talk about the topology of DNA, the double strands of which can behave a bit like knotted ribbons. Enzymes called topoisomerases help untangle them.

In another sense, though, mathematics has something in common with neckties: it’s just something we like to do, whether it’s functional or not. Sometimes humans pursue things for the simple reason that they’re complicated, or counterintuitive, or beautiful. The study of tie knots can be as whimsical and elaborate as the knots themselves, and that may be enough. “I’m interested in it because I’m interested in fun and weird things,” Vejdemo-Johansson told me.

The Desiderata

Two months ago, I decided to teach myself the Trinity knot. I opened my closet and picked a striped tie, which I hoped would accentuate the knot’s unique design. Then I stood in front of a mirror, my laptop a few steps away, and started to imitate an online tutorial. On my first few attempts, I fumbled. I wasn’t even sure that I had succeeded until I tugged and tidied the knot.

A couple of times, I tried inventing a knot. There was a lot of fiddling and fudging; it’s difficult to say where one move ends and another begins. In this sense, ties seem more like Play-Doh than Legos. Yet a bit of tightening or loosening can suddenly resolve a messy arrangement into symmetry. The process reminded me of the way that mathematicians talk about their research. Sometimes, you start with a destination and need to figure out the way there; other times, you start with a mere notion, and you wind up somewhere unexpected. You lose the path, backtrack, and find it again. When you make your final move, you hold something new in your hands: a nice, tight proof.

In October, I wore the Trinity to a gala honoring women in science, conservation, and exploration. I was surprised when no one mentioned it the entire evening, even when I told two people at my table that I was writing an article about tie knots. But when I walked into my apartment building, around eleven that evening, I found Mocka sitting behind the security desk. He grinned. “Look who’s wearing the Trinity!” he said.

I told Mocka about the lack of fanfare surrounding my tie. “There’s a subtlety to the Trinity,” he said. Sometimes tie knots inspire subconscious reactions. Sometimes they make you feel confident for the sole reason that you know you’re wearing one.

My Trinity was especially subtle given my black-and-blue-striped tie, he pointed out; I should have worn a solid color, or more contrasting stripes. “Tricks of the trade,” he said. Then he leaned over and gently loosened the loops of my tie. This way, “it pops out,” he said. I looked in a mirror, and it did. I pointed out that tieknotology was not just topology, but also psychology. “Distinction is one of those things that’s very interesting,” Mocka said.

Another resident, a young woman, walked in, and Mocka retrieved a package for her. I told her that he had designed over a thousand tie knots. Over a million, he interjected.

“What’s your favorite one, though?” she asked him.

The Jawbreaker, he said, and dropped his jaw to illustrate what tends to happen when people encounter it.

“O.K., I have to see it,” the woman said.

Mocka pulled up a picture on his laptop. There was a lot going on, but, at the top of the knot, the tie blade, facing backward, poked upward through the fabric loop that normally holds the tail in place. The knot looked like an elaborately folded napkin in a ring.

“Oh, my God! How long does that take?” the woman said.

“If you know how to do it, it takes less than two minutes,” he replied.

She scrolled through Mocka’s tie-knot collection on his laptop, glimpsing the wide world beyond the four-in-hand. “I didn’t even know that this was an option,” she said. By the time she stepped into the elevator, she was musing about creating her own knot. ♦

https://www.newyorker.com/culture/annals-of-inquiry/the-man-who-invented-fifteen-hundred-necktie-knots