MM-CAHF · Combinatorial aspects of Heegaard Floer homology for knots and links
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2021-09-01 → 2023-08-31
- EU contribution
- €139,851
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Combinatorial aspects of Heegaard Floer homology for knots and links
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.
Data: CORDIS, © European Union
Project objective
The action's goal is to achieve major advances in Heegaard Floer homology for knots and links. Heegaard Floer homology is a package of powerful invariants for 3-manifolds, and knots and links inside them. Introduced two decades ago, it is now a major research area in low-dimensional topology. To a knot or link in the 3-sphere, together with extra data called `decoration', Heegaard Floer homology associates a bigraded vector space which determines key topological properties of such a knot or link, such as its Alexander polynomial and its Seifert genus. Moreover, given a (decorated) link cobordism between two links, there is a linear map induced between their Heegaard Floer homology. The original definition of Heegaard Floer homology is based on counting pseudo-holomorphic curves in symplectic manifolds, but there exist combinatorial reformulations of the vector spaces associated to decorated knots and links.The proposal consists of three major projects:1) Give a combinatorial reformulation of the Heegaard Floer cobordism maps, to make their computation algorithmic, by extending existing combinatorial definitions of the vector spaces associated to decorated knots and links.2) Extend the most efficient combinatorial reformulation, namely the Kauffman-states functor, from decorated knots to decorated links.3) Define a combinatorial Heegaard Floer invariant for partially decorated links, for which attempts to give an analytic definition seems unfeasible.
Original text from CORDIS.
Participants
- HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET · BudapestCoordinatorHungary
Links
- View on CORDIS
- DOI: 10.3030/893282
- https://www.renyi.hu/en/overview/mm-cafh-expanding-the-combinatorial-formulations-of-heegaard-floer-homology
Data: CORDIS, © European Union