Danica Kosanović

Max-Planck Institute für Mathematik
Office B27
Vivatsgasse 7, Bonn, DE 53111

I am a final 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, see my CV or 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.

Connect-sum with Borromean rings

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. Moreover, 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.

Borromean rings isotopy

Milnor Invariants Learning Seminar (July/August 2019)

Ben Ruppik and I are organising 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!

Peter's and Aru's class on 4-manifolds (Winter Semester 2018)

I was giving tutorials for this class. Here is the page with the class notes and homework assignments.

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: Bing

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.


Embedding calculus invariants for classical knots are surjective. To appear soon.

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.

Space of gropes and the embedding calculus. In preparation.

Joint with Yuqing Shi and Peter Teichner.

On Borromean link families in all dimensions. In preparation.

Joint with Peter Teichner.

Research talks and posters

Knots map onto components of the embedding calculus tower @ Spaces of Embeddings: Connections and Applications, Banff International Research Station, Canada
A gong show talk @ Workshop on 4-manifolds, MPIM Bonn
A gong show talk @ Knots and Braids in Norway (KaBiN), Trondheim
A geometric approach to embedding calculus @ Utrecht Geometry Center Seminar
Инваријанте чворова и конфигурациони простори @ Mathematical Institute SANU, Belgrade, slides (in Serbian)
Revisiting the Arf invariant @ Topology Seminar, MPIM Bonn
Extended evaluation maps from knots to the embedding tower @ Manifolds Workshop (part of Homotopy Harnessing Higher Structures Trimester) at Isaac Newton Institute, Cambridge, slides
Knot theory meets homotopy theory @ IMPRS Seminar, MPIM Bonn, slides
Grope cobordism and the embedding tower for knots @ ICM 2018 Satellite Conference: Braid Groups, Configuration Spaces and Homotopy Theory, in Salvador, Brazil
Feb 2018
Poster A homotopy theoretic approach to finite type knot invariants @ Winter Braids, CIRM, Luminy, France

Expository talks

Hairy graphs and mapping spaces between little disk operads @ Hot Topic Seminar, MPIM
Milnor invariants and Whitney towers @ Milnor Invariants Seminar, MPIM
July 2019
Introduction to Milnor link invariants and relation to Massey products @ Milnor Invariants Learning Seminar, MPI
May 2019
Formality of little disks operads @ IMPRS seminar, MPIM
Sep/Oct 2018
Two talks about the paper of Ihara on automorphisms of pure sphere braid group @ GT learning seminar, MPIM
Apr/May 2018
Two talks on perturbative quantization and Chern-Simons theory for knots @ BV learning seminar, MPIM
Complex oriented cohomology theories @ Peter’s Seminar in Berkeley
Universal Knot Invariants @ The Chinese University of Hong Kong
How to draw a smooth 4−manifold? @ IMPRS seminar, MPIM
A categorical approach to quantum knot invariants @ Topology Seminar, MPIM
A survey of Witten-Reshetikhin-Turaev invariants of 3-manifolds @ Special Topology Seminar, MPIM
Topological reincarnations of the Arf invariant @ Cambridge Junior Geometry Tea Seminar, Cambridge, UK
Topological reincarnations of the Arf invariant @ Berkeley seminar