Research

My research is mainly on low-dimensional topology, in particular, open book decompositions in the study of 3-manifolds, knot theory, and contact geometry. This implies studying mapping class groups of compact surfaces with boundary. My thesis is now available here. My papers can be seen below (along with some questions I have about them), with the link on the title leading to the (freely available) arXiv version.

I also collaborate with Peter Feller, Lukas Lewark, Isacco Nonino, and Eric Stenhede.

The support genus does not increase under contact connected sum (joint with Eric Stenhede).

It is unknown whether there exist contact structures with support genus greater than one. In the recent problem list K3, one of the avenues proposed to attack this problem is to consider connected sums of contact manifolds of support genus one, and try to show additivity of support genus under connnected sum.

We show this strategy does not work by proving that the support genus of the connected sum of contact structures is at most the maximum of the support genera of the summands. The key input is the fact that we may always perform a Murasugi sum of open books with resulting genus the maximum of the genera of the summands. See on the right an example with two surfaces of genus one, summed to give a surface of genus one (naturally, with more boundary components).

Questions: Can this inequality be strict for tight contact manifolds? (It always is if one of the summands is overtwisted and the other one has nonzero support genus). Is the support norm additive under connected sum?

2. Non-fibered strongly quasipositive links and tightness (joint with Isacco Nonino).

Building on the previous paper (see immediately below), we associate, to an incompressible Seifert surface, a partial open book, and thus a contact manifold (possibly with boundary). We know by Hedden that a fibered link is strongly quasipositive if and only if its associated open book supports the standard tight contact structure.

We prove that non-fibered strongly quasipositive links also support tight contact structures in this new sense, but the converse is not true, see for example the figure on the right for a (non-fibered) knot that is not quasipositive but supports a tight contact structure. We also ask the following:

Question: Does the contact structure induced by a strongly quasipositive link have nonvanishing Heegaard Floer contact invariant, like in the fibered case? (I -strongly- suspect the answer is YES).

3. Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links (joint with Peter Feller and Lukas Lewark).

We introduce the notion of partial monodromy of an incompressible surface in a 3-manifold as a map defined on (a subset of) the arc set of the surface. We prove a composition formula of these partial monodromies under Murasugi sums, analogous to Gabai's result for fiber surfaces. The idea is as follows. For a Murasugi sum of (M_1, S_1) and (M_2,S_2), given a product disk we can intersect it with the individual manifolds to obtain collections of product disks in the summands, see the figure on the right for a schematic picture.

Using this, we prove an analog of the primeness criterion from Homogeneous braids are visually prime (see below), which we use to establish primeness of a large class of links (similarly to the previous paper, this class contains arborescent links). This allows us to prove Cromwell's conjecture for so-called alternative links, a class which strictly contains all previous results.

Along the way, we prove that strongly quasipositive surfaces are right-veering, and our proof allows us to recover Honda, Kazez, and Matic's characterization of tight contact structures in terms of right-veering open books, both in the classical and the partial open book case.

I gave a talk about this paper at the conference Bienal de la RSME 2026, here are the slides.

Question: Is Cromwell's conjecture true for any family bigger than alternative diagrams (for example, homogeneous diagrams)?

4. Homogeneous braids are visually prime (joint with Peter Feller and Lukas Lewark). Published by Journal of Topology, you can see the published version here.

When is it possible to tell from a link diagram whether the link represented is prime? More than 30 years ago, Cromwell proved that this happens for positive braids, and conjectured that it is also true for diagrams that produce a minimal genus Seifert surface when Seifert's algorithm is applied. The full conjecture is false by Stoimenow, but it is not known which is the most general class of diagrams for which it holds.

We resolve Cromwell's conjecture for all braid closures. We also show primeness of a large class of fibered links, which we call trees of open books. This class contains fibered arborescent links.

We are currently trying to generalise the result to other families. The main tool is a criterion for primeness under Murasugi sums. (Update: the paper is now out! See above).

I gave a talk at K-OS about this paper, here are the slides. A recording of the talk is also available here.

Question: If L is a prime link in S^3, and we plumb a trefoil to it in a non-trivial way, is the resulting link still prime? (Note: in general 3-manifolds the answer is NO, although using the figure-eight instead of the trefoil the answer becomes YES).

5. Detecting right-veering diffeomorphisms. Published in Algebraic&Geometric Topology, you can see the published version here.

The right-veering property is important for distinguishing tight contact structures form overtwisted ones. However it is often difficult to detect. This paper gives a combinatorial way of showing whether an open book is right-veering.

I am currently trying to tie this result to work of Baldwin, Ni, and Sivek that relates the right-veering property to an invariant in Knot Floer homology.

Question: Can we make the algorithm any more efficient?

6. On binding sums of contact manifolds.

The binding sum is an operation on contact manifolds similar to the connect sum. However, its behaviour is not as well understood as the connect sum case. We would like to see when certain properties (such as tightness and fillability) are preserved under this operation.

In this paper we provide an explicit computation of the vanishing of the contact class for an infinite family of binding sums whose summands are Stein fillable, showing that many properties of contact manifolds are not necessarily preserved under the binding sum. Along the way we show that contact manifolds with Giroux torsion have spectral order at most 2.

Questions: Do contact manifolds with Giroux torsion have spectral order 1? Can we show analogous results for binding sums when the summands have more than 2 boundary components?

7. Humanity's Last Exam. Published by Nature, you can see the published version here.

I took part in the project Humanity's Last Exam, which created a dataset of questions to serve as an academic benchmark for Large Language Models (LLMs), since lately the models tend to achieve over 90% on previous benchmarks. The dataset consists of 2,700 questions across over a hundred subjects. This dataset proved significantly more challenging for the models than existing benchmarks.

Consider a once-punctured torus. Take the product of the (positive) Dehn twists along the standard homology generators, and finally take its 9th power (see the image on the right). What is the fractional Dehn twist coefficient of the resulting mapping class? If you can answer this question, congratulations! You can do better than existing AI (although this might be obsolete by the time you read it. Update: ChatGPT is now able to answer this question).

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