Maxim Karev.
Floer homology of a link with a trivial component.
Heegaard-Floer homology was born as a 3-manifolds invariant,
which was defined through holomorphic disks counting and Heegaard diagrams.
Later, an analogious construction was used for defining
an invariant for null-homologous knots in a closed, oriented 3-manifold.
Quite recently {MOS} a remarkable way to compute this invariant for
knots in 3-sphere using combinatorial methods only was found.
This approach to Heegaard-Floer homology was developed in {MOST}.
Let $L$ be a link in 3-sphere and let $L^\bigcirc$ be its
\textit{extension}, i.e. the link obtained from $L$ by adding an unknotted and
unlinked component. The aim of this talk is to evaluate Heegaard-Floer
homology of an extended link $L^\bigcirc$ through the Heegaard-Floer
homology of $L$.
We don't know if Heegard-Floer homology has the functoriality property, i.e.
is it true that a smooth cobordism between links induces Heegaard-Floer
homology of theese links morphism, depending on the isotopy type of this
cobordism only. It is not hard to understand that there exists a smooth
cobordism between $L$ and $L^\bigcirc$ consisting of a cylinder and a
semisphere. We hope our result could help to solve the functoriality
problem.
References:
{MOS} C.~Manolescu, P.S.~Ozsv\`ath, S.~Sarkar. A
Combinatorial Description of Knot Floer Homology, preprint,
arXiv:math.GT/0607691.
{MOST} C.~Manolescu, P.S.~Ozsv\`ath, Z.~Szab\`o,
D.P.~Thurston. On Combinatorial Link Floer Homology, preprint,
arXiv:math.GT/0610559.