On a sunny summer day in Montauk, Ipek Irgit was in the familiar situation of not knowing what she was doing with her life. It was 2012, and she was 34 years old. Earlier in the year, after having her heart broken, she had taken a vacation in Brazil, visiting Rio de Janeiro and the beach towns of Bahia. Now she was on the beach on Long Island, wearing a handmade-looking bikini with crochet and exposed elastic straps, trying to figure out what to do and who to be.

It had been 11 unsettled years since Ms. Irgit moved from Turkey to New York, with $300 in her pocket and a vague dream of making a life in one of the city’s creative industries. She became part of adowntown class of trendsetters who all seemed to have a purpose. It was an expensive existence, especially for Ms. Irgit, who, without the cushion of a trust fund, bounced from job to job. She was a waitress and a bartender; she worked for a fashion company as an assistant to a merchandiser and in the wholesale department. Nothing stuck.

On the Montauk sand, Ms. Irgit was playing paddle ball with a friend, the restaurateur Serge Becker, when he complimented her striking bikini. As Ms. Irgit would later put it, this was the spark that motivated her to start a business. Why not create more of these distinctive bathing suits? Why not become a bikini-maker?

Ms. Irgit looked at the bathing suit and saw untapped “superpowers.” It was bright, colorful and