## Danica Kosanović

Max-Planck Institut für Mathematik
Office B27
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, embedding calculus, operads, graph complexes.

For more details, see my CV, these slides 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, 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 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.

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 $ev_n\colon\mathcal{K}\to\pi_0\mathsf{T}_n$ 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 $\pi_0$ 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 below hints 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.

### Milnor Invariants Learning Seminar (July & November 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. 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:

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

### Preprints

Space of gropes and the embedding calculus. In preparation.

Joint with Yuqing Shi and Peter Teichner.

A geometric approach to the embedding calculus knot invariants.

A preliminary draft available here. Last edit: January 14, 2020.

Abstract. We prove that the invariants $ev_n\colon\pi_0\mathrm{Emb}_\partial(I,I^3)\to \pi_0\mathsf{T}_n$ of classical (long) knots arising from the Goodwillie-Weiss embedding calculus are surjective, which completes a missing case in the Goodwillie-Klein connectivity estimates and confirms a part of the conjecture by Budney-Conant-Koytcheff-Sinha that these are universal Vassiliev invariants over $\mathbb{Z}$. Crucial ingredients are our geometric understanding of $\mathsf{T}_n$, which we develop in general for $\mathrm{Emb}_\partial(I,M)$ where $M$ is any manifold with boundary, and an explicit interpretation of $ev_n$ when $\dim(M)=3$ from \cite{KST}, using capped grope cobordisms.

Actually, we prove a more general result: the first possibly non-vanishing embedding calculus invariant of a knot which is grope cobordant to the unknot is the equivalence class of the underlying tree of the grope. As another corollary, we confirm that these invariants are universal Vassiliev invariants over $\mathbb{Q}$, which factor configuration space integrals through the tower.

On Borromean link families in all dimensions. In preparation.

Joint with Peter Teichner.

### Upcoming talks

30.1.2019
Knot invariants from homotopy theory @ Topology Seminar Bochum

### Research talks and posters

20.1.2019
Knot theory meets the embedding calculus @ Copenhagen Algebra/Topology Seminar
16.1.2019
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

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