Danica's webpage

Danica Kosanović

C = zz in pizza, Ć = ci in ciabatta

LAGA- Laboratoire analyse, géométrie et applications
Université Sorbonne Paris Nord (Paris 13)
99 avenue Jean-baptiste Clément
93430 Villetaneuse



I hold a postdoc position at LAGA (Paris 13) working with Geoffroy Horel, and funded by FSMP. In September 2021 I will move to ETH Zürich as a Hermann Weyl Instructor.

My interests include knot theory, 4-manifolds, Goodwillie-Weiss embedding calculus, operads, graph complexes. For more details, see this introduction or the slides from the public talk of my defense, or have a look at the tabs on the left.

I have obtained my PhD degree from the University of Bonn, working at the Max-Planck Insitut für Mathematik under the supervision of Peter Teichner. Previously I have studied in Belgrade (Serbia) and Cambridge (UK).

My partner Mihajlo Cekić is also a mathematician.

News and Upcoming talks

I'm organizing a learning seminar, here is the webpage.

3.12.2020 Online @ Théorie des groupes, LAMFA Université d'Amiens
4.12.2020 Online Colloquium @ LAGA Paris 13
13.1.2021 Online @ Higher Structures & Field Theory Seminar

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, TQFT’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 studying finite type invariants of knots and their relation to the Goodwillie-Weiss embedding calculus. Below I give a short introduction into these topics.

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 different story is the Goodwillie-Weiss embedding calculus. When applied to the embedding functor of long knots \(\mathcal{K}\) in the 3-space it yields a tower of spaces \(\mathsf{T}_n\) together with evaluation maps \(ev_n\colon\mathcal{K}\to\mathsf{T}_n\). These spaces turn out to be very interesting. 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 \(\pi_0\) 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 \(\pi_0\mathsf{T}_n\).

Therefore, the two stories should not be so separate after all. One unifying perspective is that of gropes. Namely, 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 depicted below hints at how this in turn relates to gropes.

In the very last picture we clearly see a genus one surface with one boundary component emerging. This will represent the bottom stage of a grope.

Borromean rings isotopy


Embedding calculus and grope cobordism of knots. See arxiv.org/abs/2010.05120.

Abstract. We show that the invariants \(ev_n\) of long knots in a \(3\)-manifold, produced by embedding calculus, are surjective for all \(n\geq1\). On one hand, this solves some of the remaining open cases of the connectivity estimates of Goodwillie and Klein, and on the other hand, it confirms one half of the conjecture by Budney, Conant, Scannell and Sinha that for classical knots \(ev_n\) are universal additive Vassiliev invariants over the integers.
We actually study long knots in any manifold of dimension at least \(3\) and develop a geometric understanding of the layers in the embedding calculus tower and their first non-trivial homotopy groups, given as certain groups of decorated trees. Moreover, in dimension \(3\) we give an explicit interpretation of \(ev_n\) using capped grope cobordisms and our joint work with Shi and Teichner.
The main theorem of the present paper says that the first possibly non-vanishing embedding calculus invariant \(ev_n\) of a knot which is grope cobordant to the unknot is precisely the equivalence class of the underlying decorated tree of the grope in the homotopy group of the layer.
As a corollary, we give a sufficient condition for the mentioned conjecture to hold over a coefficient group. By recent results of Boavida de Brito and Horel this is fulfilled for the rationals, and for the \(p\)-adic integers in a range, confirming that the embedding calculus invariants are universal rational additive Vassiliev invariants, factoring configuration space integrals.

In preparation

On certain homotopy groups of spaces of embedded arcs and circles. Email me if you would like see a copy.

Abstract. For the space of embedded arcs (long knots) in manifolds of dimension \(d\geq4\) we describe the lowest homotopy group which potentially distinguishes it from the space of immersions. To this end we both collect some existing results and also describe explicit generators of these groups. For \(d=3\) we obtain invariants of isotopy classes of knots, related to Vassiliev type \(\leq1\). We also discuss embedded circles, answering a question posed by Arone and Szymik: in a simply connected \(4\)-manifold the fundamental group of the space of embedded circles is the same as that of the space of immersed circles.

Classifying disks with duals in a 4-manifold. Coming soon, joint with Peter Teichner.

Spaces of gropes and the embedding calculus. Work in progress, joint with Yuqing Shi and Peter Teichner.

On Borromean link families in all dimensions. Work in progress, joint with Peter Teichner.


A geometric approach to the embedding calculus knot invariants. PhD Thesis. Download.

Research talks

26.11.2020 Online Knot invariants from homotopy theory @ Séminaire AGATA, Université de Montpellier, beamer slides
17.11.2020 Online Knot invariants from homotopy theory @ Warwick algebraic topology seminar
2.11.2020 Online Knot invariants from homotopy theory @ G&T Seminar Glasgow
16.10.2020 Knot invariants from homotopy theory @ Université de Lille
31.7.2020 Online Embedding calculus for knot spaces @ Oberwolfach Workshop Topologie
29.5.2020 Online Knot invariants from homotopy theory @ Topological Quantum Field Theory Seminar, Instituto Superior Técnico, Lisboa, video
21.4.2020 Online Knot invariants from homotopy theory @ jointly Séminaire de l'équipe Topologie Algébrique, LAGA, Paris 13 and Séminaire de Topologie, IMJ-PRG, Paris 7
20.2.2020 A geometric approach to the embedding calculus @ Oberwolfach Workshop Low-dimensional Topology
30.1.2020 Knot invariants from homotopy theory @ Topology Seminar Bochum
20.1.2020 Knot theory meets the embedding calculus @ Copenhagen Algebra/Topology Seminar
16.1.2020 Нове технике у теорији утапања (New techniques in the theory of embeddings) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade
2.12.2019 Knot theory meets the embedding calculus @ MPIM Topology Seminar, Bonn
16.10.2019 Knots map onto components of the embedding calculus tower @ Spaces of Embeddings: Connections and Applications, Banff International Research Station, Canada
16.9.2019 A gong show talk @ Workshop on 4-manifolds, MPIM Bonn
13.5.2019 A gong show talk @ Knots and Braids in Norway (KaBiN), Trondheim
7.5.2019 A geometric approach to embedding calculus @ Utrecht Geometry Center Seminar
25.12.2018 Инваријанте чворова и конфигурациони простори (Knot invariants and configuration spaces) @ Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, slides (in Serbian)
17.12.2018 Revisiting the Arf invariant @ Topology Seminar, MPIM Bonn
6.12.2018 Extended evaluation maps from knots to the embedding tower @ Manifolds Workshop (part of Homotopy Harnessing Higher Structures Trimester) at Isaac Newton Institute, Cambridge
28.11.2018 Knot theory meets homotopy theory @ IMPRS Seminar, MPIM Bonn, slides
24.7.2018 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

13.2.2020 On the punctured knots model for embedding spaces @ Configuration Categories Learning Seminar (Online)
19.12.2019 On link maps @ Mojito’s Seminar (Online)
13.2.2020 On the paper by Bundey-Gabai about knotted 3-balls @ Online Student Seminar, notes
19.12.2019 Watanabe's counting formula for classes in Diff(S^4) @ Hot Topic Seminar, MPIM
5.11.2019 Milnor invariants and Whitney towers @ Milnor Invariants Learning Seminar, MPIM
July 2019 Introduction to Milnor link invariants and relation to Massey products @ Milnor Invariants Learning Seminar, MPIM
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
22.3.2018 Complex oriented cohomology theories @ Peter’s Seminar in Berkeley
06.12.2017 Universal Knot Invariants @ The Chinese University of Hong Kong
15.11.2017 How to draw a smooth 4−manifold? @ IMPRS seminar, MPIM
25.09.2017 A categorical approach to quantum knot invariants @ Topology Seminar, MPIM
04.08.2017 A survey of Witten-Reshetikhin-Turaev invariants of 3-manifolds @ Special Topology Seminar, MPIM
02.06.2017 Topological reincarnations of the Arf invariant @ Cambridge Junior Geometry Tea Seminar, Cambridge, UK
23.03.2017 Topological reincarnations of the Arf invariant @ Berkeley seminar

Building Bridges Seminar: Invariants of embedding spaces

(Winter 2020)

See the tab seminar.

Working Group on Grothendieck-Teichmüller Group (and Applications)

(Winter 2020)

Geoffroy Horel and Bruno Vallette are organizing a learning seminar at Paris 13. See here.

Milnor Invariants Learning Seminar

(July & November 2019)

Ben Ruppik and I were organising a series of talks on Milnor invariants. Ben made a cool website which contains our notes and references.

Peter's and Aru's class on 4-manifolds

(Winter 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.