Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.
problem Determining which knots can yield Seifert fibered homology spheres.
method Utilized Heegaard Floer and Knot Floer homology, along with the mapping cone formula, to find obstructions.
result Showed that genus one knots cannot yield Seifert fibered homology spheres with six or more singular fibers.
Paper connects homologically fibered links to solutions of equations.
problem Existence of homologically fibered links in 3-manifolds.
method Interprets existence in terms of first homology group and torsion linking form, or surgery diagrams.
result Computes hc for specific 3-manifolds.
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.
problem Calculating Witten-Reshetikhin-Turaev invariants for Seifert fibered homology 3-spheres.
method Explicit modular transformation formulas of homological blocks.
result New proof of Witten asymptotic conjecture for Seifert fibered homology 3-spheres.
New knot homologies detect non-fibered knots, expanding on previous results.
problem Detecting non-fibered knots using knot homologies.
method Developed new knot homologies (knot Floer, Khovanov, HOMFLY) and proved a technical theorem.
result New homologies detect infinitely many non-fibered knots, expanding on previous results.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Every lens space has a simple knot with specific properties.
problem Finding specific knots in lens spaces.
method Using Chebotarev density theorem and number theory.
result Proves existence of genus one homologically fibered knots in lens spaces.
Defines a new symplectic Khovanov homology for links in fibered 3-manifolds.
problem No specific problem stated, but deals with Khovanov homology for links in fibered 3-manifolds.
method Defines a symplectic Khovanov type homology for a transverse link in a fibered closed 3-manifold with an auxiliary loop.
result Conjectural combinatorial dgas for surface categories, higher-dimensional analogs of strands algebras.
Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.
problem Computing the first homology group of the Milnor fiber boundary for generic hyperplane arrangements.
method Analyzing the Milnor fiber boundary for hyperplane arrangements in C^3.
result Affirmative answer to the conjecture of Suciu and example of arrangements with non-trivial torsion.
We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In pa…
Floer homology linked to Milnor fibers for certain singularities.
problem Understanding Floer homology of singularities using Milnor fibers.
method Combining Floer theory with Milnor fibers, using combinatorial formulas.
result Equality of Casson invariants in Donaldson and Seiberg-Witten theories.
Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.
problem Linking fibered classes in 3-manifolds via Alexander polynomials.
method Applies twisted Alexander polynomials to fibered classes in ribbon homology cobordisms.
result Fibered classes of Y+ map to those of Y−. Collects properties of Seifert homology spheres for concordance studies.
problem Understanding concordance properties of fibers in Seifert homology spheres.
method Using Dehn surgery along Seifert fibers.
result Includes various properties in one place for concordance studies.
Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certa…
Homological blocks match Witten-Reshetikhin-Turaev invariants for Seifert fibered 3-spheres.
problem Matching homological blocks with WRT invariants for specific 3-manifolds.
method Developed an asymptotic formula and vanishing result of coefficients.
result Radial limits of homological blocks match Witten-Reshetikhin-Turaev invariants.
Study shows Seifert fibered spaces don't bound rational homology balls.
problem Understanding when Seifert fibered spaces bound rational homology balls.
method Analyzes Seifert fibered spaces with different conditions and orientations.
result Characterizes conditions for Seifert fibered spaces to bound rational homology balls.
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …
Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.
problem Understanding which Seifert fibered spaces can be boundaries of symplectic rational homology balls.
method Analyzes convex boundaries and Lagrangian disk fillings of Legendrian knots.
result Strong restrictions on which Seifert fibered spaces can bound symplectic rational homology balls.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
Study calculates Heegaard Floer homology for Seifert fibered 3-manifolds.
problem Computing Heegaard Floer homology for Seifert fibered 3-manifolds.
method Combinatorial models inspired by lattice homology.
result Connected Floer homology determines local equivalence class of ι-complex. The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.
problem Understanding the unique connected sum of lens spaces formed by a special knot in the Poincaré sphere.
method Analyzing the Seifert fibering and Dehn surgery of the Poincaré homology sphere.
result The only knot in the Poincaré sphere with a surgery to a connected sum of more than two lens spaces is the one mentioned.
In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each pri…
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic…
Icosidodecahedral arrangement has torsions in homology, and is a K(π,1).
problem Identifying arrangements with torsions in homology.
method Proving the icosidodecahedral arrangement is a K(π,1).
result The icosidodecahedral arrangement is a K(π,1).
New stability result for Milnor fibers of pure braid groups.
problem Understanding the stable homology of Milnor fibers.
method Representation stability in the context of Milnor fibers of pure braid groups.
result Computed stable integral homology of Milnor fibers.
Positive braid knots have simple knot Floer homology.
problem Computing knot Floer homology for positive braids.
method Computed the next-to-top term of knot Floer homology.
result Rank is 1 for any prime positive braid knot.
Floer homology detects right-veering monodromy in fibered knots.
problem Detecting right-veering monodromy in fibered knots.
method Using Heegaard Floer homology and symplectic Floer homology.
result Knot Floer homology characterizes tight contact structures.
Study shows quotient group is infinitely generated.
problem Understanding the structure of homology cobordism groups.
method Proved using Seifert fibered spaces.
result Quotient group is infinitely generated.
We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold…
Proves nontrivial knot Floer homology for fibered knots.
problem Determining the knot Floer homology of fibered knots.
method Uses Alexander grading and 3-manifold surgeries.
result Knot Floer homology is nontrivial in the next-to-top Alexander grading for fibered knots.
New proof shows L-space knots are fibered and have specific properties.
problem Proving L-space knots are fibered and have strong quasipositive properties.
method Conceptually simpler proof using Heegaard Floer homology, translated to instanton Floer homology.
result Generalization of L-space knot properties to other Floer homologies.
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
New models found for guts of nearly fibered knots.
problem Characterizing nearly fibered knots topologically.
method Provided three models for the guts of nearly fibered knots.
result Nearly fibered condition can be purely topologically characterized.
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
Knot Floer homology reveals fixed points of monodromy.
problem Understanding fixed points of monodromy for fibered knots.
method Knot Floer homology and monodromy analysis.
result Monodromy of a fibered knot has at most rank minus one fixed points.
Study on mod 2 Betti numbers of complex hyperplane arrangements and Milnor fiber homology.
problem Determining mod 2 Betti numbers of complex hyperplane arrangement complements.
method Combining the mod 2 Aomoto complex and the transfer long exact sequence.
result First homology of Milnor fiber has non-trivial 2-torsion for the icosidodecahedral arrangement.
Proves additivity of Casson-Seiberg-Witten invariant for a specific 4-manifold.
problem Additivity of Casson-Seiberg-Witten invariant for integral homology S1imesS3. method Proves additivity under fiber sum along embedded curves and embedded tori.
result Establishes the 4-dimensional analogue of the Casson invariant additivity. The study computes invariants of satellite knots using bordered Floer homology.
problem Computing invariants of satellite knots with specific patterns.
method Using bordered Floer homology and the immersed curve interpretation of the bordered pairing theorem.
result Satellites with thin fibered companions or specific patterns have thin knot Floer homology.
New generalization of knot Floer homology rank conjecture.
problem Rank of knot Floer homology in the next-to-top Alexander grading.
method Independent generalization to a family of knots with specific conditions.
result Generalization verified for a family of knots with Seifert surfaces satisfying certain conditions.
This paper shows that only finitely many knots can be ribbon concordant to any given knot.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Using knot Floer homology and bordered Heegaard Floer homology, the paper proves an inequality relating the knot Floer homology of a satellite knot and its companion.
result Every knot in S3 has only finitely many fibered predecessors under ribbon concordance. The paper examines Seifert surgeries on knots via torsion and invariant.
problem Analyzing Seifert surgeries on knots using torsion and invariant.
method Using Reidemeister torsion and Casson-Walker-Lescop invariant.
result Conditions for the number of singular fibers in Seifert fibered spaces.
We show that if the fundamental group of the complement of a rationally homologically fibered knot in a rational homology 3-sphere is bi-orderable, then its Alexander polynomial has at least one positive real root. Our argument can be applied for a finitely generated group which is an HNN extension with certain propert…
Quantum invariants of Seifert fibered homology spheres are resummated and related to classical Chern-Simons values.
problem Resummation and classification of quantum invariants for Seifert fibered homology spheres.
method Analysis of q-series and Ohtsuki series, asymptotic expansions, Borel transform, and classification of moduli spaces.
result The limit of the resummation equals the WRT quantum invariant and classifies components of the moduli space.
Instanton homology detects 2-torsion in fibered knots.
problem Detecting 2-torsion in instanton homology for fibered knots.
method Using sutured instanton theory to derive a formula for I♯(Y,K;C) and comparing dimensions. result Proves the presence of 2-torsion in instanton homology for null-homologous fibered knots.
Maps Heegaard Floer homology to Hecke algebras for surfaces.
problem Connecting Heegaard Floer homology with Hecke algebras.
method Constructs a map from Heegaard Floer homology to Hecke algebras.
result Shows the map is an isomorphism of algebras.