the webpage of Danica Kosanović

Danica Kosanović

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ETH Zürich, Forschungsinstitut für Mathematik
Rämistrasse 101, HG GO 68.2
8092 Zürich, Switzerland



I am a Hermann Weyl Instructor at the Departement of Mathematics at ETH Zürich (Swiss Federal Institute of Technology in Zurich). My mentors are Peter Feller and Thomas Willwacher.

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

Before coming to ETH in 2021, I was a FSMP postdoc at Paris 13 (Université Sorbonne Paris Nord), working with Geoffroy Horel. I obtained my PhD degree in September 2020 from the University of Bonn (Germany), working at the Max-Planck Insitut für Mathematik under the supervision of Peter Teichner. Previously, I studied in Belgrade (Serbia) and Cambridge (UK).

My partner Mihajlo Cekić is also a mathematician.


May 2023 I co-organised the workshop "Interactions of Low-dimensional Topology and Quantum Field Theory"
July 2023 My new preprint "Knotted families of (long) knots from graspers" is on arXiv, see
December 2023 The preprint "Homotopy versus isotopy: 2-spheres in 5-manifolds" of Rob Schneiderman and Peter Teichner and myself is on arXiv, see


Spring 2024 I'll give lectures on Topology of Manifolds, see the tab "Teaching".

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, can merge together to give even more insight about low-dimensional manifolds.

In my thesis I studied finite type knot invariants and their relation to the Goodwillie-Weiss embedding calculus.


Embedding calculus and grope cobordism of knots

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary 3-manifold. This solves some remaining open cases of Goodwillie–Klein–Weiss connectivity estimates, and at the same time confirms one half of the conjecture that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. In addition, we give a sufficient condition for this conjecture to hold over a coefficient group, which is by recent results of Boavida de Brito and Horel fulfilled for the rationals and for the $p$-adic integers in a range. Therefore, embedding calculus invariants are strictly more powerful than the Kontsevich integral.

Furthermore, our work shows they are more computable as well. Namely, the main theorem computes the first possibly non-vanishing invariant $ev_n$ of a knot which is grope cobordant to the unknot to be precisely equal to the equivalence class of the underlying decorated tree of the grope in the associated graph complex. Actually, our techniques apply beyond dimension $3$, offering a description of the layers in embedding calculus for long knots in a manifold of any dimension, and suggesting that certain generalised gropes realise the corresponding graph complex classes.

A space level light bulb theorem in all dimensions
~ Joint with Peter Teichner ~
accepted for publication in Commentarii Mathematici Helvetici

Given a $d$-dimensional manifold $M$ and a knotted sphere $s\colon\mathbb{S}^{k-1}\hookrightarrow\partial M$ with $1\leq k\leq d$, for which there exists a framed dual sphere $G\colon\mathbb{S}^{d-k}\hookrightarrow\partial M$, we show that the space of neat embeddings $\mathbb{D}^k\hookrightarrow M$ with boundary $s$ can be delooped by the space of neatly embedded $(k-1)$-disks, with a normal vector field, in the $d$-manifold obtained from $M$ by attaching a handle to $G$. This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree $d-2k$. In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

A new approach to light bulb tricks: Disks in 4-manifolds
~ Joint with Peter Teichner ~
accepted for publication in Duke Mathematical Journal

For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings $\mathbb{D}^2\hookrightarrow M$ with boundary $k$, using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of $M$.

On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant
accepted for publication in Transactions of the American Mathematical Society

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M of dimension $d\geq 4$. In particular, if M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M, answering a question posed by Arone and Szymik. The case $d = 3$ gives isotopy invariants of knots in a 3-manifold, that are universal of Vassiliev type $\leq 1$, and reduce to Schneiderman’s concordance invariant.

Knotted families from graspers
accepted for publication in Journal of Topology

For any smooth manifold $M$ of dimension $d\geq4$ we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $M$, in every degree that is a multiple of $d−3$, and show that they are detected in the Taylor tower of Goodwillie and Weiss.

Homotopy versus isotopy: 2-spheres in 5-manifolds
~ Joint with Rob Schneiderman and Peter Teichner ~

In this note we give a complete obstruction for two homotopic embeddings of a 2-sphere into a 5-manifold to be isotopic. The results are new even though the methods are classical, the main tool being the elimination of double points via a level preserving Whitney move in codimension~3. Moreover, we discuss how this recovers a particular case of a result of Dax on metastable homotopy groups of embedding spaces. It follows that ``homotopy implies isotopy'' for 2-spheres in simply-connected 5-manifolds and for 2-spheres admitting algebraic dual 3-spheres.


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

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.

Embedding calculus of Goodwillie and Weiss is another homotopy-theoretic approach to spaces of embeddings. When applied to the embedding functor of long knots $\mathcal{K}$ in the 3-space it yields a tower of spaces $T_n$ together with evaluation maps $ev_n\colon K\to T_n$. 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 $\pi_0T_n$.

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

Research talks

29.11.2023 Knotted families from graspers, BIRS Workshop Spaces of Manifolds: Algebraic and Geometric Approaches, Banffhandwritten slides (~14MB)
7.9.2023 Knotted families from graspers, Conference Swiss Knots 2023, Regensburg
9.5.2023 2-knots with light bulbs and knotted families of arcs, University of Neuchâtel, Colloquium
5.5.2023 - Online Homotopy groups of some embedding spaces Part II, Kansas State University Topology Seminar
28.4.2023 - Online Homotopy groups of some embedding spaces, Kansas State University Topology Seminar
12.3.2023 - Online A new approach to light bulb tricks, Online Conference "Interactions of 3- & 4-dimensional Topology", webpage
17.11.2022 2-knots and knotted families of arcs, University of Zurich, room Y27-H-26, at 10:30
04.11.2022 - Online A new approach to light bulb tricks @ BIRS Workshop Topology in Dimension 4.5, Banff
24.-27.10.2022 Homotopy groups of some embedding spaces @ Réunion annuelle du GDR, 24-27 October 2022, Université de Nantes, webpage, slides
14.9.2022 2-knots and knotted families of arcs @ DMV (German Mathematical Society) Annual Meeting, Berlin, beamer slides
16.7.2022 2-knots and knotted families of arcs @ AMS-EMS-SMF Congress of Mathematics, 18-22 July, 2022, Grenoble
25.3.2022 - Online A light bulb theorem for disks @ Algebraic Geometry and Differential Topology Seminar, Alfréd Rényi Institute of Mathematics, Budapest
24.2.2022 Smooth embeddings and their families @ Durham Geometry and Topology Seminar
16.2.2022 Smooth embeddings and their families @ Cambridge Differential Geometry and Topology Seminar
11.2.2022 - Online Smooth embeddings and their families @ UL Lafayette Topology Seminar
7.10.2021 - Online A light bulb theorem for disks @ Princeton Topology Seminar
6.10.2021 Light bulbs in 4-manifolds @ ETH Zürich Geometrie Seminar
21.9.2021 - Online A light bulb theorem for disks @ University of Virginia Geometry Seminar
17.8.2021 Учворени дискови у четири димензије @ Workshop on Symplectic Topology, University of Belgrade
9.7.2021 - Online Higher homotopy groups in low dimensional topology @ Young Topologists Meeting, beamer slides
11.6.2021 - Online A light bulb theorem for disks @ Georgia Topology Conference, beamer slides
21.4.2021 - Online Knotted families of arcs @ Münster Topology Seminar
15.3.2021 - Online Knotted families of arcs @ MIT Topology Seminar
13.1.2021 - Online Knot invariants from homotopy theory @ Higher Structures & Field Theory Seminar
4.12.2020 - Online Knot invariants from homotopy theory @ Colloquium LAGA Paris 13
3.12.2020 - Online Knot invariants from homotopy theory @ Théorie des groupes, LAMFA Université d'Amiens
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 @ BIRS Workshop Spaces of Embeddings: Connections and Applications, Banff
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

25.5.2022 Freedman's work on 4-manifolds @ Actualité des mathématiques, Neuchâtel
6.5.2021 On a theorem of Kontsevich and Conant-Vogtmann @ Groupes de Grothendieck-Teichmüller et applications
11.3.2021 Chord diagram invariants of tangles @ Groupes de Grothendieck-Teichmüller et applications, notes
13.2.2020 - Online On the punctured knots model for embedding spaces @ Configuration Categories Learning Seminar
19.12.2019 - Online On link maps @ Mojito’s Seminar
13.2.2020 - Online 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

Topology of Manifolds

(Spring 2024)


Lecture Notes.


Smooth manifolds.
Topological manifolds.