Quantum invariants, knot concordance and unknotting

点击翻译按钮获取中文摘要

My research is in low-dimensional topology and knot theory. This is a very active domain of pure mathematics, which has seen extraordinary development over the last decades, such as the advent of categorified quantum invariants. It is a multifaceted domain in which many influences mingle. Knots are naturally related to concepts from 3-dimensional topology, such as hyperbolic geometry or contact geometry. One can see knots from the perspective of algebraic topology; this gives a connection to group theory (via the fundamental group) and the theory of quadratic forms (via homology). Knot diagrams - projections of a knot to the plane - are essentially decorated plane graphs and allow for the application of combinatorial and graph theoretic techniques in knot theory. They also lead to a connection to quantum algebra. Aside from their relation to 3-manifolds, knots also enjoy close ties with 4-manifolds, and the dichotomy of the smooth and topological category of 4-manifolds manifests itself on the level of knots.My principal ambition is to understand the bridges between these different parts of knot theory. Here are three programmatic questions guiding my research:(1) What topological information is contained in the Seifert form?(2) What geometrical information is contained in quantum invariants?(3) What do knot diagrams reveal about unknotting and surfaces in 3-space?Let me elaborate on each of those goals. For (1), I am using a mixture of 3-dimensional and algebraic techniques relating to quadratic forms, ultimately building on Freedman’s work. I have worked on related questions for some time, and developed an effective upper bound for the topological slice genus. My next step will be to characterise completely algebraically the genus of topological surfaces in 4-space whose complement has infinite cyclic fundamental group. It is rather rare that such a topological quantity can completely be determined in an algebraic way.Regarding (2), I will profit from years of experience with quantum invariants, in particular Khovanov-Rozansky homologies. It is an upshot of my work (combining methods from homological algebra with computer calculation) that the smooth concordance information in those homologies is much richer than had been generally assumed, and potentially strong enough to detect odd torsion in the concordance group. Categorified quantum invariants form a young and very dynamic subject; my work is directed at one of the central questions of the subject, namely the invariants' geometrical interpretation.Goal (3) touches on a topic on which I have just begun to work. Nevertheless, using braid manipulation and graph theoretic methods I am planning to obtain first results soon: I am going to show which canonical surfaces are quasipositive, that almost positive knots are strongly quasipositive, and that the unknotting number of positive fibred knots agrees with their genus. Each of these statements resolves an open conjecture.