Abstract: Knot theory has found some applications in biology and chemistry with the study of knotted structures such as DNA, polymers and other molecular and biological structures. Among those latter however, there are far more complex structures than simple curves like knots. We would like to study some of them, namely the 3-periodic tangles, by providing a new diagrammatic description based on diagrams of usual knots. I will present how those diagrams are created, how do they work, how do they describe fully a 3-periodic tangle and its equivalence class, and present some examples.

Just how knotted is a knot? One way to measure this is to find the minimal number of crossing changes needed to transform it into the unknot. This so called unknotting number is one of the most studied knot invariants but it still remains unknown for many knots and there is no general algorithm known to determine it. Thanks to the results of Rudolph and others, the unknotting number of braid positive knots is known to coincide with its Seifert genus, which can be easily computed. Furthermore, there is a more general class of knots known as positive fibered knots for which it is currently unknown whether the unknotting number coincides with their Seifert genus. In an attempt to find a counterexample to this assertion, we discovered a knot for which no existing method for determining the unknotting number was found to be effective.

This is based on joint work with M. Kegel, L. Lewark, N. Manikandan, F. Misev, and M. Silvero.

Abstract: We discuss extensions of classical knot theory to the setting of 1-dimensional submanifolds in R³ with one direction of periodicity. Commonly known examples of this are braids and chains, but these are just a small subset of this much broader class of objects. We present generalizations of classical polynomial and hyperbolic invariants to this setting and study their properties, using a projection to the quotient space. A particular focus is on different notions of equivalence for these knottings beyond the standard ambient isotopy equivalence. Finally, we show how to generate knot tables for these structures algorithmically, and discuss how such classification efforts can inspire new theorems.

Abstract: The crosscap number of a knot is the non-orientable counterpart of its genus. It is defined as the minimum of one minus the Euler characteristic of S, taken over all non-orientable surfaces S bounding the knot. Computing the crosscap number of a knot is tricky, since normal surface theory - the usual tool to prove computability of problems in 3-manifold topology, does not deliver the answer "out-of-the-box".

In this talk, I will review the strengths and weaknesses of normal surface theory, focusing on why we need to work to obtain an algorithm to compute the crosscap number. I will then explain the theorem stating that an algorithm due to Burton and Ozlen can be used to give us the answer.

This is joint work with Jaco, Rubinstein, and Tillmann.