Danica's webpage

# Danica Kosanović

Pronounce c as zz in pizza, and ć as ci in ciabatta. Typeset the letter ć as \'{c} in LaTeX.

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

kosanovic[at]math[dot]univ-paris13[dot]fr

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, knotted surfaces in 4-manifolds, homotopy types of embedding spaces, 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 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 studied in Belgrade (Serbia) and Cambridge (UK).

My partner Mihajlo Cekić is also a mathematician.

### News and Upcoming talks

I'm organizing a "Building Bridges" learning seminar, here is its webpage.

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 studied finite type knot invariants and their relation to the Goodwillie-Weiss embedding calculus. Here is a short introduction to 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.

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.

### Preprints

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

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.

### In preparation

On certain homotopy groups of spaces of embedded arcs and circles. To appear soon.

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.

A space level light bulb theorem for disks. To appear soon, joint with Peter Teichner.

We describe the homotopy type of the space of embedded $k$-disks in a $d$-manifold $M$ with fixed boundary in $\partial M$, for which there exists a geometric dual sphere, in terms of the space of embedded $(k-1)$-disks in the manifold obtained from $M$ by attaching a handle to the dual sphere. As a consequence, for $k=2$ and $d=4$ we can use the Dax invariant of knotted arcs to completely classify such $2$-disks in a $4$-manifold up to isotopy, answering questions posed in \cite{Gabai-disks}. Moreover, we recover the classification of homotopic $2$-spheres up to isotopy, with the Dax invariant reducing to the Freedman-Quinn invariant of \cite{Schneiderman-Teichner}. This lifts the light bulb theorem of \cite{Gabai-spheres, Schneiderman-Teichner} to the level of spaces.

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.

### Thesis

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

### Research talks

 21.4.2021 Online Knotted families of arcs @ Münster Topology Seminar beamer slides 15.3.2021 Online @ MIT Topology Seminar 13.1.2021 Online @ 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 @ 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

 11.3.2021 Chord diagram invariants of tangles @ Groupes de Grothendieck-Teichmüller et applications, notes 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.

### The class on 4-manifolds by Peter Teichner and Aru Ray

(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:

3. Here are solutions to some of the homework exercises we didn't have time to cover in the tutorials.