Combinatorial aspects of Heegaard Floer homology for knots and links

Project Information

MM-CAHF

Grant agreement ID: 893282

DOI

10.3030/893282

Project closed

EC signature date 20 March 2020

Start date 1 September 2021

End date 31 August 2023

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 139 850,88

EU contribution

€ 139 850,88

Coordinated by

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Hungary

Periodic Reporting for period 1 - MM-CAHF (Combinatorial aspects of Heegaard Floer homology for knots and links)

Reporting period: 2021-09-01 to 2023-08-31

Summary of the context and overall objectives of the project

The action's research topic is the study of knots and link in the mathematical field of topology. If we consider a circle in the familiar three-dimensional space, this can be knotted, and in fact in many different ways. For example, the DNA of some bacteria is a molecule that appears in the form of a knotted circle. From a mathematical perspective, we say that such a molecule describes a "knot", or a "link" if there are multiple components. The main goal of the action is to understand and develop tools to study knots and links from the mathematical perspective of the field of topology. The action moved in two different directions: on one hand, we expanded our knowledge of the current mathematical invariants used to study these objects, and on the other hand we explored their applications to the study of topology in dimension four.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

A significant portion of the work revolved around an algorithmically computable invariant for knots and links known as knot Floer homology, which has been defining this field of research in the past two decades. Our work produced new connections between the rank of the knot Floer homology of a given knot (satisfying certain properties) and an important topological invariant, the fusion number of the knots. A further algebraic structure, the homological action, will be studied in a forthcoming paper. Connections between knots and 4-dimensional spaces were also explored, in two different directions: the study of a maps induced by cobordisms, and surfaces with boundary a certain given link.
The results of the action have been disseminated through 3 conferences and 14 seminar talks.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project advanced our understanding of the interactions between knots and 4-dimensional topology, with a special focus on some 4-dimensional manifolds (complex projective planes, K3 surfaces). The results of the projects opened up new interesting directions of research and have sparked the interest of the low-dimensional topology community.
On a different note, the connections between the rank of knot Floer homology and Khovanov homology and some invariants of knots open up new questions for the future.
Finally, cobordism maps between knots were studied during the action, and more work is planned on the topic for the next few years.

A blackboard with thoughts