During his introductory remarks, Dr. Reis had a vexing encounter with an intertwined microphone cord. “This is a good example of why we really care about this topic,” he said.

Dr. Reis grapples with the likes of long-overhand shoelace knotting, climbing knots, basket weaving, surgical sutures, and how to pass on the art of surgical-knot craftsmanship to robots. During the session, his lab mates described how they had used a CT scan to probe the internal structure of knot filaments and the friction that arises where filaments touch. After the meeting, Dr. Matsumoto sent him home with some of her swatches.

Derek Moulton, of the University of Oxford, mentioned variants of sailor’s knots, DNA and protein knots, and worms that tie themselves into knots in order to minimize dehydration. He went on discuss “whether a knotted filament with zero points of self-contact may be realized physically.” That is, can a knot exist wherein none of its crossings touch? (It can; try it at home with a strip of paper, or a cord.)

And Thomas Plumb-Reyes, an applied physicist at Harvard, presented his research on “Detangling Hair” to a standing-room-only audience.

“What is going on in tangled hair?” he asked. “What is the optimal combing strategy?”

Shashank Markande, a Ph.D. student working with Dr. Matsumoto, reported on their stitch classification work so far. Together, they had derived a conjecture: All knittable stitches must be ribbon knots. (A ribbon knot is a very technical tangle.) And they pondered the corollary: Are all ribbon knots knittable?

Back in February, Mr. Markande (who started knitting only recently for the sake of science) thought he’d found an example of an unknittable ribbon knot, using a knots-and-links software program called SnapPy. He sent Dr. Matsumoto a text message with a sketch: “Tell me if this can be knitted?”

Dr. Matsumoto was just heading out for a run, and by the time she returned, having manipulated the yarn every which way in her head, she had worked out an answer. “I think that can be knitted,” she texted back. When Mr. Markande pressed her on how, she added: “It’s knittable by our rules, but it isn’t trivial to do with needles.”