Danica KosanovićMax-Planck Institute für Mathematik
Vivatsgasse 7, Bonn, DE 53111
I am a second year PhD student at the Max-Planck Insitut für Mathematik, under supervision of Peter Teichner. My studies are funded by the Max Planck Society, and I am a member of the International Max Planck Research School on Moduli Spaces (IMPRS). I have previously studied in Belgrade (Serbia) and Cambridge (UK).
My interests include knot theory, 4-manifolds, embedding calculus, operads, graph complexes.
For more details, have a look at the tabs on the left.
I like thinking about knots, 4-manifolds, surfaces inside, and in general about topology in low dimensions! However, I also believe that formalism and tools of higher topology, i.e. homotopy theory, higher categories, TFT’s, operads, as well as combinatorics of Feynman diagrams and configuration spaces, should merge together to give even more insight about low-dimensional manifolds.
In my thesis I am looking at finite type invariants of knots and their relation to the embedding calculus of Goodwillie and Weiss. Below I give a short introduction into this topic; more about my research can be found in the tab ''Papers and Talks''.
Finite type invariants (often called Gusarov-Vassiliev, or just Vassiliev, invariants) give a certain filtration on the set of all invariants by their type. A dual point of view is, however, more geometric: there is a filtration on the monoid of knots itself, which arises by looking at a certain sequence of n-equivalence relations on knots. Then the n-th term of the filtration is comprised of knots which are n-equivalent to the unknot.
For example, two knots are 1-equivalent if they can be related by a sequence of crossing changes. This means that the first term in the filtration is equal to the whole monoid of knots! To get an idea about 2-equivalence, take a look at the operation on the left - grab some three strands of a knot and connect-sum them with the Borromean rings.
A completely separate story can be told about the Goodwillie-Weiss embedding calculus applied to the embedding functor of long knots in the 3-space. This yields a tower of spaces together with evaluation maps evn:K→π0Tn from the space of knots to the levels of the tower. These spaces turn out to be very interesting and mysterious. For example, they can be shown to be double loop spaces of the mapping spaces between some (truncated) operads. Hence, their components form an abelian group and the evaluation map from knots gives a map on π0 which turns out to be - a finite type invariant. It is conjectured to be universal such, in other words, the group of knots modulo relation of n-equivalence is isomorphic to π0Tn.
Hence, two stories are not so separate after all. Moroever, the trivalent vertices appearing in the diagrams for finite type theory (originating in quantum Chern-Simons theory) correspond to the Borromean rings, and the isotopy below hints how this in turn relates the n-equivalence with gropes. Namely, in the very last picture we clearly see a genus one punctured surface, which will represent the bottom stage of a grope.
Ben Ruppik and I organizing a series of talks on Milnor invariants. Ben made a cool website which contains our notes and references so far, as well as a rough plan for the second part - this should happen at the end of August or in September. Stay tuned!
1. Here are the level sets of the Boy's surface.
2. Here is the proof that Bing double of any knot is a boundary link:
3. Here are solutions to some of the homework exercises we didn't have time to cover in the tutorials.
4. See also interior and boundary twists.
[work in progress] Embedding calculus invariants for classical knots are surjective
It has been conjectured that the evaluation maps coming from the embedding calculus are universal knot invariants of finite type. One half of this statement is that these maps are surjective on components. I am working on completing my proof of this, building on a result previously obtained jointly with Yuqing Shi and Peter Teichner.