Imagine taking a tangled extension cord and plugging it into itself, so that it forms a closed loop. Can it be untied without unplugging the end? If not, can you prove it?
Topology is the study of mathematical spaces up to smooth deformations — stretching, twisting, bending — that do not break the space. Like many problems in topology, it is intuitively obvious that a closed, knotted loop can’t be untied, but backing up this intuition with mathematical rigor can be much more complicated.
Knot theory — the study of knotted loops in 3-dimensional space — was developed in the 19th century, motivated by the mistaken proposition of Lord Kelvin that every atom was actually a knot in the ether, a ubiquitous substance composing all matter, and different types of atoms corresponded to different types of knots. This theory was debunked in the early 20th century, but knot theory persisted on its own, first as a mathematical curiosity, then emerging as one of the centerpieces in low-dimensional topology over the last 50 years.
Ethereal connections notwithstanding, knot theory can be applied to any number of physical contexts, including biology and chemistry. Perhaps most notably, Edward Witten earned a Fields Medal (the only physicist to do so) in part by discovering deep connections between knot invariants and fundamental concepts in physics.
An invariant of a knot is a function that takes a knot as input and outputs another mathematical structure, such as an integer, a polynomial or a group. Invariants are, in a way, like a blood type — if two blood samples have different types, they must have come from different individuals. Similarly, if two knotted loops have different invariants, they must be distinct knots.
Various faculty members in the department have been conducting research to better understand these structures, including Mark Brittenham, who has been publishing papers in knot theory for nearly 20 years. Most recently, Brittenham is working to better understand the unknotting number of a knot, which in some sense measures how “close” a particular knot is to being unknotted. He has implemented an exhaustive computer search to determine new unknotting numbers, with great success; his work is now included in KnotInfo, a comprehensive database of knots and their invariants maintained by Indiana University and used by researchers all over the world.
Alex Zupan, who joined the faculty in the fall of 2015, is studying another classical knot invariant, the bridge number of a knot. Every knot K can be cut into two collections of unknotted arcs by an embedded plane in 3-dimensional space, and the bridge number of K is number of arcs in a minimal such decomposition. Much of Zupan’s work as a postdoctoral researcher at the University of Texas at Austin involved understanding decompositions coming from other, more complicated surfaces, determining properties about a sequence of these invariants, called the bridge spectrum. Following the completion of that work, he adapted the notion of bridge number to dimension four, finding a new invariant of knotted surfaces in four-dimensional space.
Zupan’s arrival neatly coincided with recent UNL graduate Nick Owad’s dissertation year (2015-2016), as Owad’s thesis work extended Zupan’s results about the bridge spectrum to new families of knots. Owad is but one in a long string of graduate students advised by Brittenham and/or Susan Hermiller and doing research in various topics in knot theory. Like Brittenham, both Melanie DeVries (Ph.D., ’13) and Anne Kerian (Ph.D., ’15) were interested in unknotting numbers of knots, with DeVries extending ideas to the broader context of virtual knots, and Kerian discovering new unknotting results for a specific knot family.
The research group has also included undergraduate students. For instance, Lucas Sabalka — who went on to earn his Ph.D. from UIUC — completed an undergraduate thesis in braid groups, a subject at the intersection of knot theory and group theory, under the direction of Hermiller and John Meakin in 2002.
In addition, the department has seen collaboration in knot theory research among faculty, with Brittenham and Hermiller writing numerous joint papers in knot theory and other topics, often with collaborators from around the world. Their most recent effort involves UNL postdoctoral researcher Timothy Susse and another noteworthy invariant, the fundamental group of a topological space. This invariant is often used to distinguish two knots, but its impact extends far beyond the realm of knot theory.
Knot theory is closely related to the study of 3-manifolds, where a 3-manifold is a topological space that locally looks like 3-dimensional real space. In September of 2016, Brittenham, Hermiller, and Susse posted a preprint proving the monumental result that if G is the fundamental group of a 3-manifold, then G has a special property known as autostackability. Being autostackable roughly means that a computer with finite memory could be programmed to decide whether a given group element in G, expressed in a complicated way, is actually the trivial element. Although certain types of 3-manifolds were previously known to have this property, this major theorem unifies groups of all 3-manifolds in a clear and elegant way.
While knot theory can be a source of many difficult and longstanding research problems, it is also a wonderfully intuitive and tangible subject, including topics suitable for graduate students, undergraduate students, and even high school students — Zupan taught a mini-course in knot theory for UNL’s All Girls/All Math program during the summer of 2016. We look forward to seeing the many exciting directions in which the study of knot theory evolves as a part of UNL’s teaching and research mission.
-Alex Zupan