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We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots.
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We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish.
We associate several invariants to a knot in an integer homology 3-sphere using $SU(2)$ singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern-Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Fr{o}yshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowkas singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the 4-sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples.
We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_isubset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $widehat{mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $widehat{mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$big|(h_infty^1-h_1^1)(h_infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)big|.$$ We also show that if splicing the complement of a knot $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori.
This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.
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