The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure.Given Heegaard splitting = U 0 ∪ Σ 1 these theories are variants the Lagrangian Floer homology g-fold symmetric product relative totally real subspaces associated and .

In [27], we introduced Floer homology theories HF -(Y, s), ∞ (Y, + t), s),and red s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ (Y ).In the present paper, give calculations and study properties of these invariants.The suggest conjectured relationship Seiberg-Witten theory.The include between Euler characteristics ± Turaev's torsion, minimal genus problem (Thurston norm), surgery exact sequences.We also some applications techniques three-manifold topology.

We prove that, like the Seiberg-Witten monopole homology, Heegaard Floer homology for a three-manifold determines its Thurston norm.As consequence, we show that knot detects genus of knot.This leads to new proofs certain results previously obtained using (in collaboration with Kronheimer and Mrowka).It also purely Morse-theoretic interpretation knot.The method proof shows canonical element associated weakly symplectically fillable contact structure is non-trivial.In particular,...

We use the knot filtration on Heegaard Floer complex to define an integer invariant tau(K) for knots. Like classical signature, this gives a homomorphism from concordance group Z. As such, it lower bounds slice genus (and hence also unknotting number) of knot; but unlike tau sharp four-ball genera torus another illustration, we calculate several ten-crossing

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere.To obtain this result, we use long exact sequence for monopole homology, together with nonvanishing theorem, which shows detects unknot.In addition, apply these techniques give information about knots admit lens space surgeries, and exhibit families of three-manifolds do not taut foliations.

In an earlier paper, we introduced a knot invariant for null-homologous K in oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. this paper investigate some properties these groups knots three-sphere. We give combinatorial description generators chain complex and their gradings. With help description, determine alternating knots, showing that special case, it depends only on signature Alexander polynomial (generalizing result Rasmussen two-bridge knots)....

Let K be a rationally null-homologous knot in three-manifold Y .We construct version of Floer homology this context, including description the obtained as Morse surgery on .As an application, we express Heegaard rational surgeries along terms filtered homotopy type invariant for .This has applications to Dehn problems knots S 3 .In different direction, use techniques developed here calculate arbitrary Seifert fibered with even first Betti number.

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian homology. In earlier paper, this was given combinatorial description with mod 2 coefficients. the present we give self-contained presentation basic properties link homology, including elementary proof its invariance. We also fix signs differentials, so that theory integer

Knot Floer homology

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces symmetric algebra appearing definition with exterior algebra. The two invariants have same reduction modulo 2, but differ over rationals. There a reduced version link whose graded Euler characteristic normalized polynomial.

In an earlier paper (math.SG/0101206), we introduced Floer homology theories associated to closed, oriented three-manifolds Y and SpinC structures. the present paper, give calculations study properties of these invariants. The suggest a conjectured relationship with Seiberg-Witten theory. include between Euler characteristics Turaev's torsion, minimal genus problem (Thurston norm), surgery exact sequences. We also some applications techniques three-manifold topology.

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in three-sphere, which takes values link homology. This can be used also construct of transverse knots.

Due to the proliferation of large language models (LLMs) and their widespread use in applications such as ChatGPT, there has been a significant increase interest AI over past year. Multiple researchers have raised question: how will be applied what areas? Programming, including generation, interpretation, analysis, documentation static program code based on promptsis one most promising fields. With GPT API, we explored new aspect this: analysis source front-end at endpoints data path. Our...

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative.These relations, together with Taubes' basic theorems on the of symplectic manifolds, are then used to prove Thom conjecture: surface fourmanifold genus-minimizing its homology class.Another corollary relations general adjunction inequality for negative four-manifolds.

Remark 1.2.• We choose an orientation of det(H 2 ≥0 (M, ∂M, R))⊗(det(H 1 R))) -1 called a homology ∂M ).Standard Mayer-Vietoris arguments show that for ) induces M γ , i.e. det, and hence orients the moduli spaces allowing us to define function SW .In statement theorem all functions etc. are computed using orientations derived from common ).• If b + (M ≥ 2, then there no chamber structures invariants.).This identification cohomology positive cones with their structures., set spin c...

This article analyzes the interplay between symplectic geometry in dimension $4$ and invariants for smooth four-manifolds constructed using holomorphic triangles introduced [20]. Specifically, we establish a nonvanishing result of four-manifolds, which leads to new proofs indecomposability theorem Thom conjecture. As application, generalize splittings along certain class three-manifolds obtained by plumbings spheres. restrictions on topology Stein fillings such three-manifolds.