Before this summer, Jacqueline Gipe knew nothing about Knot Theory, a field of mathematics. Now, she is knee deep in Knot Theory research uncovering proofs relating to chord diagrams with the help of her mentor and AQ mathematics professor, Dr. Michael McDaniel.

Gipe is a junior at AQ and is double-majoring in Spanish and math. She is also taking education classes and hopes to one day teach math at the secondary level.

Gipe was chosen to carry out this research project based on her answers to a set of questions that McDaniel proposed. “I put up a geometry question in Wikispaces from a previous summer research paper which had not been published. So I knew nobody could simply Google the answer,” said McDaniel. “Then I sent out an email to all our declared majors who had taken Topics in Geometry who were not seniors, requesting their solutions. I got three replies and Jackie's was the best.”

McDaniel has his doctorate degree in Knot Theory. He came up with the original idea for the research project, which is based on a question that he worked on seven years ago but had fallen just short of the answer. “Last spring, I was home and a new attack [for the project] came to mind while I was playing with the ferrets. It turned out that this new attack worked and Jackie and I can prove some theorems about the finite-type knot invariants,” said McDaniel.

Gipe and McDaniel’s research deals with chord diagrams, which are circles that have chords going through them. These diagrams represent singular knots (knots that intersect), which differs from a regular knot because regular knots simply have crossings where the knot goes over or under itself, instead of an intersection.

“Dr. McDaniel and I are working with a specific structure of chord diagrams, all the chord diagrams that come from the permutations of the closed Jacobi diagram,” said Gipe. “So it's basically a wheel, and the permutations are the different variations of this diagram we can have by changing the order of the chords coming off of it. We are trying to prove that all of the chord diagrams that come from the closed Jacobi diagram are invertible. A chord diagram is invertible if we can switch the orientation on the chord diagram and we still get the same chord diagram back.”

Gipe notes that because this question is not yet proved, it is even more difficult because she knows that she cannot just look at the back of a textbook for a hint. Yet this is also what makes the journey that more exciting and rewarding.

“There are many failures during research where you get an idea and think it's great and then it turns out there's a flaw in your proof. Dr. McDaniel has shown me that it doesn't matter if your idea ended up being wrong, it's all about working toward your goal and getting further than you, or anyone else, has gotten before on a difficult problem,” said Gipe.

Gipe is determined to keep working through the difficult subject matter. She finds it easier to tackle this subject because of McDaniel’s help and expertise in the area. “Dr. McDaniel has been extremely helpful and patient in helping me understand the basic concepts of Knot Theory. He's also shown me what it's like to do research in mathematics.”

Conducting research can be tedious and time consuming but Gipe and McDaniel have found an outlet after a hard days work in McDaniel’s pet ferrets. McDaniel said the ferrets give he and Jackie “a much-needed source of laughs when we're up to our ears in math.”

McDaniel thinks highly of Gipe and is happy to be working with a dedicated student such as herself. “I gave her a brief summary of math algorithms which were completely new to her on our first day and she dove in with gusto,” said McDaniel. “In a world where so many people are uncreative, tedious television addicts, spending a happy ten hours with paper and pen does not happen much. Jackie creates patterns of polynomials using known algorithms, teases further structures out of these patterns and then proves the structures hold in general, which contributes new facts to the world of mathematics. Not many undergraduates at any college do this. Aquinas College is fortunate to have the Mohler-Thompson grants AND the students strong enough to maximize the opportunity.”

McDaniel is confident in the duo’s work so far. “Of course, until our paper passes the referee's study, we shouldn't claim that we have our theorems done,” said McDaniel. ”We have been writing it up and the proofs look solid.”

Gipe and McDaniel will continue researching until the end of July. Gipe will present their research at the Mohler-Thompson Research Presentations at Aquinas in the fall of 2013.