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.”
