# Danica Kosanović

(C is pronounced as zz in pizza, Ć as ci in ciabatta)

Max-Planck Institut für Mathematik
Vivatsgasse 7, Bonn, DE 53111
danica[at]mpim-bonn[dot]mpg[dot]de

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, Goodwillie-Weiss embedding calculus, operads, graph complexes.

For more details, see my CV, these slides or have a look at the tabs on the left.

### News

I have submitted my thesis on May 29, 2020.

From March till August 2020 I will be visiting IHES.

In October 2020 I will start a postdoc position at Université Paris 13 working with Geoffroy Horel, funded by FSMP.

7.2020
Topologie @ Oberwolfach

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.

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.

### In preparation

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

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.

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. The submitted version.

Abstract. In this thesis we consider two homotopy theoretic approaches to the study of spaces of knots: the theory of finite type invariants of Vassiliev and the embedding calculus of Goodwillie and Weiss, and address connections between them.

Our results confirm that the knot invariants $ev_n$ produced by the embedding calculus for (long) knots in a 3-manifold M are surjective for all n ≥ 1. On one hand, this solves certain remaining open cases of the connectivity estimates of Goodwillie and Klein, and on the other hand, confirms a part of the conjecture by Budney, Conant, Scannell and Sinha that for the case $M =I^3$ of classical knots $ev_n$ are universal additive Vassiliev invariants over $\mathbb{Z}$.

There are two crucial ingredients for this result. Firstly, we study the so-called Taylor tower of the embedding calculus more generally for long knots in any manifold with $dim(M) \geq 3$ and develop a geometric understanding of its layers (fibres between two consecutive spaces in the tower). In particular, we describe their first non-vanishing homotopy groups in terms of groups of decorated trees. Secondly, we give an explicit interpretation of $ev_n$ when $dim(M)\geq 3$ using capped grope cobordisms. These objects were introduced by Conant and Teichner in a geometric approach to the finite type theory, but turn out to exactly describe certain points in the layers.

Our main theorem then states that the first possibly non-vanishing embedding calculus invariant 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. The surjectvity of $ev_n$ onto the components of the Taylor tower follows from this immediately.

As another corollary we obtain a sufficient condition for the mentioned conjecture to hold over a certain coefficient group A. Namely, it is enough that the spectral sequence for the homotopy groups of the Taylor tower, tensored with A, collapses along the diagonal. In particular, such a collapse result is known for $A = \mathbb{Q}$, confirming that the embedding calculus invariants are universal rational additive Vassiliev invariants, and that they factor configuration space integrals through the Taylor tower. It also follows that they are universal over the p-adic integers in a range depending on the prime p, using recent results of Boavida de Brito and Horel.

Moreover, the surjectivity of $ev_n$ implies that any two group structures on the path components of the tower, which are compatible with the connected sum of knots, must agree.

Finally, we also discuss the geometric approach to the finite type theory in terms of the Gusarov– Habiro filtration of the set of isotopy classes of knots in a 3-manifold. We extend some known techniques to prove that the associated graded quotients of this filtration are abelian groups, and study the map which relates these groups to certain graph complexes.

### Research talks and posters

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, slides
20.2.2020
A geometric approach to the embedding calculus @ Oberwolfach Workshop on 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, slides
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

### 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 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:

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