Knot Your Average Math Problem: Knot Invariants and Alexander’s Polynomials

Journalist: Emma Goshgarian | Editor: Joseph Serpico | Photographer & Videographer: Margaret Balliet

It seems that when asked how they feel about math, more often than not students will react with discomfort, even outright horror. Although many of us struggle with calculations and graphs, there is an elegant beauty and awe inherently connected with advanced mathematics; fundamentally we understand the foundational quality of math as it underlies our beloved fields, from science to music. Some students skip the messy relationship with math altogether and embrace it as a language that makes perfect sense to them.

Claire Bodemann ’18, is one of those students. A math major and self-professed “math nerd,” Bodemann gave a talk on February 17-18 to the Florida Section of the Mathematical Association of America, one of nine undergraduates among two dozen professors and graduate students. She delivered a presentation concerning her research with invariants of knots and Alexander polynomials.

Bodemann has been researching invariants in knots, the same kind of knots you use to tie your shoes, except often smaller and more complicated. Bodemann says that the applications of knots are endless and significant: “DNA can be knotted and instead of cutting it in research and breaking up the fragments on accident, they’re trying to figure out how to analyze them mathematically.”

This in itself is fascinating, but I wanted to know how knots can be mathematical. If anything, they seem like a physics experiment. In laymen’s terms, knots get their different equations from invariants, to which Bodemann especially devoted her attention, and these equations are integral in reducing- “untangling”—the knot. Bodemann uses the analogy of tying your shoes, explaining that “at the end, you fuse [the laces] together,” so that the knot is continuous. At that point, an invariant can develop; an invariant is “the value or property of a knot that can distinguish its equivalence to another knot.” Admitting that it’s a complex idea, Bodemann offers another analogy to me: “We’re both girls, and then there are boys,” so that type of characteristic difference is actually part of a mathematical distinction that leads to different equations. These equations are known as Alexander’s polynomials, and they define a knot.

So why are invariants important? Bodemann tells me that she would “take a knot and find its invariant, and then I could take the same knot and add a twist and see if I got the same polynomial.” Basically, she’s trying to make knots less complicated to see if they’re at all related. Back to the shoelace analogy, whether you use the ‘bunny-ears’ or the ‘circle’ technique, you get the same knot. Bodemann is trying to take away all the twists and see if the fundamental knots are the same.

Her talk was well-received; there is increasing research into this branch of topology, a field of pure mathematics. Bodemann plans to pursue research to this effect for her remaining year at Eckerd, and even more so in graduate school. Her enthusiasm for mathematics is evident; when asked if she truly loved math, she replied with an emphatic yes. For students like Bodemann, math is a literal and a figurative knot, which she finds truly satisfying to untangle.