Periodic Reporting for period 1 - MM-CAHF (Combinatorial aspects of Heegaard Floer homology for knots and links)
Berichtszeitraum: 2021-09-01 bis 2023-08-31
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.
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.
The results of the action have been disseminated through 3 conferences and 14 seminar talks.
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.
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.