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CORDIS

Combinatorial aspects of Heegaard Floer homology for knots and links

Project description

Expanding the combinatorial formulations of Heegaard Floer homology

Heegaard Floer homology is a powerful invariant for studying key objects in low-dimensional topology, especially knots and links in 3-manifolds. Given a cobordism between two links, there is an induced map between their Heegaard Floer homologies. The Heegaard Floer homology for knots and links in the three-sphere is algorithmically computable using certain combinatorial approaches. The EU-funded MM-CAHF project has a two-fold objective: expand the combinatorial formulations of Heegaard Floer homology to encompass a wider class of objects, and extend such formulations to cobordism maps, in order to make these maps algorithmically computable too.

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.

Coordinator

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Net EU contribution
€ 139 850,88
Address
REALTANODA STREET 13-15
1053 Budapest
Hungary

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Region
Közép-Magyarország Budapest Budapest
Activity type
Other
Links
Total cost
€ 139 850,88