Presenter(s): Nathaniel Schieber − Mathematics
Faculty Mentor(s): Robert Lipshitz
Oral Session 2S
Research Area: Knot Theory (Mathematics)
Funding: Mercer Family Foundation Scholarship, UO Department of Mathematics Juilfs Scholarship
Knot theory is exactly what it sounds like. It studies how pieces of string can be tied around themselves and around each other. From this tangible starting point, a wealth of abstract mathematics has arisen. My research in knot theory has had two main goals: to study a specific tangle and to classify tangles up to small complexity. Both have centered on encoding and manipulating the three dimensional geometry of knots within a computer program. The specific tangle I am studying is known as Krebes’s Tangle, named for the mathematician who first asked if it were possible to connect the ends of this specific tangle to the ends of a second tangle in order to form a single un-knotted circle of string. My method in approaching this question has been computational, writing code which generates random tangles, accomplishes the gluing process, and then computes a knot invariant known as the Alexander polynomial. In order to classify tangles, my code takes these randomly generated tangles and organizes them into equivalence classes based on what are known as quantum invariants. Both projects are still on–going.
Knot theory has found applications across mathematics as well as in data analysis and DNA research. However, the software for generating and manipulating generic knots directly has remained relatively limited. Along with working toward generalizing the Alexander polynomial, my work adds to the computational resources available to mathematicians studying knots. I hope it to prove of experimental benefit.