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Quantum invariants, knot concordance and unknotting

Applicant Dr. Lukas Lewark
Subject Area Mathematics
Term from 2019 to 2024
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 412851057
 
Final Report Year 2023

Final Report Abstract

Geometric topology is an area of pure mathematics. It is the study of geometric objects called manifolds and their interrelations. The mathematical theory of knots is at the center of geometric topology, since knots may be used to describe and analyze in particular manifolds of dimensions 3 and 4 – arguably the most difficult and most interesting dimensions for geometric topologists. Knots may be investigated with a variety of very different tools. The objective of this Emmy Noether Project was to establish ties between some of those different perspectives on knot theory, while paying special attention to geometrical applications. One approach to knot theory is diagrammatic: using combinatorial techniques on knot diagrams, i.e. drawings of knots in the 2 dimensional plane. Usually, the goal is then to derive from the properties of the diagram of a knot geometric consequences for the knot itself. Let me just highlight two such results of this Project. Firstly, in an article by Feller-Lewark-Lobb, the following is proven: if a diagram is almost positive (has only one negative crossing), then the corresponding knot is strongly quasipositive (this is a geometric property, related to algebraic geometry and the theory of braids). This resolved an open conjecture. Secondly, by Baader-Lewark-Misev-Truöl, for diagrams arising as the closure of a braid on three strands, there is a simple algebraic characterization of those corresponding knots for which the minimal genus of a surface with boundary that knot is equal in 3-dimensional and 4-dimensional space. One of the tools used in this Project is Khovanov homology, a so-called categorified quantum invariant, which was introduced in 1999 and is thus relatively young. This Project revealed two new geometric applications of Khovanov homology: one (Iltgen-Lewark-Marino) is a lower bound for the proper rational unknotting number, generalizing work by Alishahi-Dowlin. The other (Lewark-Zibrowius) is a new concordance invariant defined via satellite operations. In view of its algebro-diagrammatic origins, the wealth of geometric information (beginning with Rasmussen’s invariant from 2004) coming from Khovanov homology is rather astonishing. In dimension 4, the theory of differentiable (“smooth”) and topological manifolds (which are less smooth, so to say) are in stark contrast. This dichotomy has been one of the driving forces of the field ever since the groundbreaking works by Donaldson and Freedman in the 1980s (for which both received a Fields medal). On the level of knots, the applications of Khovanov homology are in the smooth category. In the topological category, a main result of this Project (Feller-Lewark) were complete algebraic/3-dimensional characterizations of the minimal genus of certain topological surfaces in 4-space. These characterizations may be seen as a quantitative version of a theorem of Freedman’s that algebraically characterizes the existence of certain topological disks in 4-space.

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