For 35 years now, the work of UB mathematics professor Bill Menasco has been wound up in knots. Once considered an esoteric pursuit, knot theory has been gaining traction in the real world. After all, those twists and turns do more than just tie our shoes—they make up our very DNA. Quantifying information about knots, and what happens when they are altered in some way, can help advance research in molecular biology, computer science and other fields.
Knot theory is a strand of geometric topology, which asks questions about an object’s essential characteristics if continuous changes are made to it—if it’s manipulated, but not cut or destroyed. How do you mathematically distinguish such changes? How do you codify that information? Knot theory asks those questions about knots in particular.
In the double helix model of DNA, there’s this spiraling of different molecular bonds that would become entangled during replication if it were just split down its axis. Nature has a way of untangling those strands— it’s where enzymes come in—and biologists want to understand that process. They’ve done experiments using electrophoresis in which magnetic fields were used to attract pieces of DNA suspended in a gel. Dependent upon how knotted they were, the pieces responded at different rates, providing a means for codifying what a particular enzyme was doing. Then the mathematicians said, “OK, you have this sequence of knots going on. If you push your techniques a little further, you ought to see this.” And the biologists went back, and, in fact, their prediction was correct. That’s mathematics at its best: It not only describes what you’re seeing—it tells you what you should be seeing.
Knot theory has relevance to robotics in the planning of configuration spaces. Imagine a manufacturing floor with a lot of automated processes, and you want this machinery to move around in such a way that it doesn’t run into itself. Picture those pathways as knots, and you can see how knot theory can be applied there too.
It’s really exploding. When I started working in knot theory, it didn’t seem like there were any real-world applications. Now, with the conjunctions it’s made with biology, physics and chemistry, it seems like you can’t do mathematics—even the very abstract kind—without it being applicable.