Studies of abstract knots

Cecilia Karlsson presented her Ph.D. in Mathematics at Uppsala University in February 2016. Thanks to a grant from the Knut and Alice Wallenberg Foundation, she will hold a postdoctoral position with Professor Yakov Eliashberg at Stanford University in Palo Alto, USA.

One can imagine a mathematical knot as a piece of rope whose ends have been glued together. There are infinitely many such knots and finding ways to distinguish them from one another is a significant challenge in knot theory. Two mathematical knots are equivalent if one can transform one into the other without resorting to cutting. Knot theory is the subject of Cecilia Karlsson’s research.

The theory first developed in the early 19th century. Later on, in the second half of the 19th century, motivated by Lord Kelvin’s conjecture that atoms were knots in the aether, mathematicians made significant contributions to knot theory. They hoped to derive the periodic table of elements by classifying knots. Before the mathematicians succeeded with their classification, physicists gave up on Kelvin’s conjecture. However, development of mathematical knot theory continued and has flourished over the last thirty years.

Mathematical knot theory can be extended in several directions, including generalizations to higher dimensions and the imposition of geometric constraints. The proposed project focuses on a special class of knots called Legendre knots. These are phenomena that emerge in the context of contact geometry which, along with the related theory of symplectic geometry, has roots in classical mechanics. Contact geometry imposes certain constraints on Legendrian knots.

Techniques from contact geometry have facilitated the discovery of new properties in Legendrian knot theory, aiding their classification. The goal of the project is further research on Legendre knots and the structures surrounding them. Over the last few years, developments in this field have been massive, and research results have contributed to other fields, including string theory in theoretical physics.