UCD Mathematicians untangle knotty problems

 

Knots are useful: they keep shoes on our feet, they anchor tents in blustery weather and they stop ships from slipping away to sea. For scientists, knots also pose particularly beautiful and interesting challenges. At UCD, mathematicians have made a breakthrough in understanding relationships between particular types of knots and other mathematical objects.

 

Recently, mathematicians Stavros Garoufalidis (Georgia Tech), Thang Le (Georgia Tech) and Don Zagier (Max Planck Institute for Mathematics) published an intriguing paper detailing more than 40 conjectural identities that connect classes of knots called alternating knots to mathematical functions called modular forms.  In 2014, Adam Keilthy worked with Dr. Robert Osburn (UCD School of Mathematics and Statistics) on a summer research project. The initial goal of the project was to look at solving some of the conjectures. Keilthy, an undergraduate Maths student at TCD, soon had a bigger goal in his sights.

 

"In the beginning I thought if we could solve one or two of these cases over the summer, then that would be great," recalls Dr. Osburn. "But a week later, Adam said he had figured out to deal with a common obstruction across the conjectures and he was able to solve all of the original  conjectures. It was a nice surprise." 

 

In 2016, Keilthy and Osburn published their findings in a 26-page paper in the Journal of Number Theory. In 2015, another of Dr. Osburn's summer interns, UCD undergraduate student Paul Beirne, took  the conjectures a step further. "The original conjectures were for alternating knots up to 8 crossings. There was one irritating knot that did not have a conjectural answer," explains Dr. Osburn.

That "question mark" knot motivated Beirne to look at what would happen if you considered all alternating knots up to 10 crossings. Another published paper arose from the work: "Paul was able to make connections to modular forms for some knots with 9 and 10 crossings, but there were a lot more question marks," explains Dr. Osburn. "The knots with 8 crossings seem to be a threshold beyond which the conjectures become less predictable."

Both students have gone on to undertake doctorates in Mathematics, Keilthy at the University of Oxford and Beirne at UCD.

Dr. Osburn sees the immense value of the UCD School of Mathematics and Statistics Summer Research Undergraduate Projects program not only for publications but also to familiarise students with the thinking processes and tools that underpin research: "It is very different from the coursework you do as an undergraduate. I think it gives students insights into mathematical research that will benefit them later in their careers."