Proves any three or more knots can form a genus-zero link in a 3-manifold.
problem Realizing any finite collection of knots as components of a genus-zero link.
method Proves the realizability of knots as components of genus-zero links in 3-manifolds, controlling pairwise linking numbers.
result Any finite collection of at least three isotopy classes of knots can form a genus-zero link in a 3-manifold, satisfying a specific condition.
Finitely many pseudo-Anosov flows without perfect fits in a 3-manifold.
problem Finite number of pseudo-Anosov flows without perfect fits in a 3-manifold.
method Analysis of veering triangulations and pseudo-Anosov flows.
result Finiteness of pseudo-Anosov flows without perfect fits.
New framework for quantum invariants of 3-manifolds using homology.
problem Quantum invariants of 3-manifolds.
method Homological construction of mapping class group representations extended to cobordisms.
result Construction of a (2+1)-dimensional TQFT assigning twisted homology to surfaces. Probabilistic pseudo knots model uncertain knot diagrams.
problem Modeling knots with unresolved crossings.
method Assign probabilities to undetermined crossings; define probabilistic equivalence and extend classical knot invariants.
result Capture uncertainty in physical, biological, and computational contexts.
An alternate proof shows how foliation extensions work in 3D spaces.
problem Continuous extension of foliations in 3D spaces.
method Uses the universal circle and properties of pseudo-Anosov flows.
result Shows how all continuous extensions organize in the boundary.
Dehn surgery on knots in 3-manifolds rarely results in surface fibered manifolds.
problem Understanding when Dehn surgery on knots in 3-manifolds results in surface fibered manifolds.
method Analyzing the fiberedness of Dehn surgeries on knots in 3-manifolds.
result Dehn surgery on knots in 3-manifolds is not surface fibered for all but a sparse set of slopes.
The paper develops a new Floer theory for 3-manifolds with involutions.
problem Constructing a Floer theory for 3-manifolds with symmetries.
method Develops a framed real monopole Floer homology for 3-manifolds with involutions and provides definitions for invariants.
result Defines Z-valued framed real Seiberg--Witten invariants for 4-manifolds with involutions. A universal branched 3-manifold W characterizes Sol 3-manifolds.
problem Characterizing Sol 3-manifolds
method Using a universal branched 3-manifold W and a regular language result A closed 3-manifold M immerses into W if and only if M admits a Sol structure. Deciding 3-manifolds fibering over the circle is in NP
problem Deciding if a 3-manifold fibres over the circle
method Show fibredness lies in NP
result Problem is NP-complete
CR singularities in 3-manifolds can be cancelled by an isotopy supported in an arbitrarily small neighborhood of a Seifert surface.
problem Canceling CR singularities in 3-manifolds
method Isotopy supported in an arbitrarily small neighborhood of a Seifert surface
result CR singularities can be cancelled
Introduces a new CR invariant for co-oriented contact structures on closed 3-manifolds
problem Distinguishing contact structures on closed 3-manifolds
method Constructs an invariant μM(ξ) associated with a contact structure ξ and open book decomposition result Shows that the first Chern classes of two tight contact structures on the 3-torus are different
New invariant distinguishes tight contact structures on 3-tori.
problem Distinguishing tight contact structures on 3-manifolds.
method Introduced a new CR invariant μM(ξ) for co-oriented contact structures on 3-manifolds. result First Chern classes of two tight contact structures on 3-torus are different.
Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.
problem Understanding PD3 groups and their properties. method Coarse generalization of Shapiro's lemma, homological isoperimetric inequalities, and Margolis's coarse homological algebra.
result Groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel are either torus or Klein-bottle bundles over S^1, or quasiisometric to Riemannian manifolds.
Rational homology ribbon cobordism defines a partial order on 3-manifolds.
problem Classifying 3-manifolds based on their homology
method Proving a partial order on homeomorphism classes of 3-manifolds
result Closed, connected, oriented 3-manifolds are partially ordered by rational homology ribbon cobordism.
New methods reveal rare epimorphisms linking 3-manifold groups to free groups.
problem Understanding which groups can be fundamental groups of 3-manifolds.
method Constructing and analyzing splitting coordinate-surjective homomorphisms.
result Splitting epimorphisms are rare and can be reduced to standard form.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.
problem Refining Alexander polynomials and bounds BNSR Σ-invariants for 3-manifolds and Kähler manifolds.
method Introduces twisted homology jump loci and uses tropical geometry to obtain bounds.
result Sharp bounds for BNSR Σ-invariants and obstructions to geometric realizability.
3-manifolds explained through geometry, proving Thurston's conjecture.
problem Thurston's conjecture about 3-manifolds.
method Explain geometrisation theorem and its proof by Perelman.
result Proved Thurston's conjecture about 3-manifolds.
New invariant from quantum algebra for 3-manifold bundles.
problem Quantum invariants of flat 2-bundles over 3-manifolds.
method From an involutory Hopf algebra graded by a crossed module, constructing a homotopy invariant via χ-colored Heegaard diagrams. result Reduces to Kuperberg invariant when bundle is trivializable.
The paper presents new algebraic structures on the 2-sphere using topological field theories.
problem Understanding algebraic structures on the 2-sphere through topological field theories.
method Defined P-monoids and L-monoids, analyzed their properties, and related them to 3-dimensional topological field theories.
result New algebraic structures (P-monoids and L-monoids) on the 2-sphere are equivalent and have strong constraints.
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
Algorithm transforms weakly negative plumbing trees to negative definite ones.
problem Transforming weakly negative plumbing trees to negative definite ones.
method Combining plumbing calculus with diagonalization algorithm to systematically eliminate positive eigenvalues.
result Explicit algorithm reduces weakly negative definite plumbing trees to negative definite ones.
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
New theorem shows embedding impossibility for certain 3-manifolds.
problem Embedding restrictions in 3-manifolds.
method Construction of hyperbolic 3-manifolds with complicated ideals.
result Existence of non-embeddable manifolds in certain 3-manifolds.
Defines real link Floer homology for specific types of links.
problem Developing a new homology theory for certain types of links.
method Combining real Heegaard Floer homology and real sutured Heegaard Floer homology, using real grid diagrams in S3. result Observes structural and property properties of strongly invertible knots.
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−. Real bordered Floer homology computes 3-manifolds with involution.
problem Computing real Heegaard Floer homology for 3-manifolds with involution.
method Using bordered Heegaard Floer algebra modules, a practical algorithm is given.
result Computes real Heegaard Floer homology for real 3-manifolds with connected fixed set.
New method calculates Thurston norm for 3-manifolds with toroidal boundaries.
problem Computing the Thurston norm for 3-manifolds with toroidal boundaries.
method maw dual graph construction and sutured manifold hierarchies.
result Explicit procedure to compute Thurston norm from hierarchies.
Refined 3D index uses surgery and gradings to distinguish 3-manifolds.
problem Distinguishing 3-manifolds and gauge theories phases.
method Dehn surgery presentation, ideal triangulation, and enhanced flavor symmetries.
result Invariance of refined index under various transformations.
Formula for computing triple-cup product from Heegaard diagrams of 3-manifolds.
problem Computing the triple-cup product invariant of 3-manifolds.
method Explicit formula from Heegaard diagrams and reduction of Turaev's homotopy intersection form.
result Triple-cup product can be recovered from Heegaard diagrams and Turaev's form.
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.
problem Determining the structure of 3-manifolds using their absolute Galois groups.
method Defined a relative absolute Galois group for 3-manifolds and used Chebotarev density properties and Hilbert ramification theory.
result Two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic.
The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
The paper classifies Morse functions on 3-manifolds with specific level sets.
problem Characterizing 3-manifolds using Morse functions with certain level sets.
method Study of Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles.
result Classification of Morse functions on specific 3-manifolds.
Study the moduli space of reducible 3-manifolds using prime decomposition.
problem Understanding the homotopy type of moduli spaces of reducible 3-manifolds.
method Construct a splitting map from BextrmDiff+(M) to BextrmDiff+(P1⊔⋯⊔Pn), yielding a prime decomposition fibre sequence. result The fibre Hg(P1,…,Pn) is a finite, connected cell complex, and the prime decomposition fibre sequence is effective for computations. Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.
problem Equivalence of two quantum field theories.
method Geometric quantization and modular categories.
result Toral Chern-Simons theory is isomorphic to Reshetikhin-Turaev theory.
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.
problem Equivalence between U(1) Chern-Simons and Reshetikhin-Turaev TQFTs. method Proof of natural isomorphism between theories for finite quadratic modules.
result Extended (2+1)-dimensional TQFTs are naturally isomorphic. The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
Simon's knot genus problem solved with 3-manifold groups.
problem If epimorphism exists between knot groups, knot genus of one is greater than or equal to the other.
method Proved conjecture linking 3-manifold groups and Thurston norms, showed locally indicable groups are Lewin groups.
result Existence of epimorphism between knot groups implies knot genus inequality.
The study examines conditions for Haken 3-manifolds and their fundamental groups.
problem Conditions for Haken 3-manifolds and their fundamental groups.
method Conditions and finite-sheeted covers of integral homology three-spheres.
result Lower bound of 4 for first Betti number of certain covers.
Algorithm finds knot friends in 3-manifolds.
problem Identifying knots that share similar 3-manifold complements.
method Developed an algorithm using SnapPy and Regina.
result Constructed a census of simple knots with friends.
Proves a property of certain 3D knots and links.
problem Understanding the fundamental groups of 3D knots and links.
method Uses profinite rigidity to analyze two-bridge links and 3-tangle Montesinos links.
result Profinite rigidity for two-bridge links and 3-tangle Montesinos links is established.
This paper uses sheaf theory to constrain knot types in clean intersections.
problem Understanding constraints on knot types in clean intersections.
method Microlocal sheaf theory and 3-manifold theory.
result Existence of a surjective homomorphism preserving longitude and meridian.
The study limits the number of cosmetic surgeries for certain knots in specific 3-manifolds.
problem Limits the number of cosmetic surgeries for knots in specific 3-manifolds.
method Rational surgery formula for Casson--Walker--Lescop invariant, constraints for null-homologous knots.
result At most two pairs of integral purely cosmetic surgeries for null-homologous knots in rational homology spheres.
Khovanov homology extended to 3-manifolds, linking tangles.
problem Extending Khovanov homology to 3-manifolds.
method Cutting 3-manifolds into intervals, constructing type D and A structures, gluing back.
result Recovering Khovanov homology of links in 3-manifolds.
Researchers debunk a generalized Property R conjecture for 2-component links.
problem The generalized Property R conjecture for 2-component links in S3. method Disproved a further generalization of the GPRC using specific counterexamples.
result 2-component framed links in S3 surging to a connected sum of homology S1imesS2's are not handleslide equivalent to a split link. Researchers lift knot coloring polynomial to Habiro ring.
problem Lifting colored Jones polynomial of knots to Habiro ring.
method Introduced new Habiro ring and map, used Alexander polynomial.
result Existence of loop expansion at roots of unity confirmed.
New formula for torsion function in 3-manifolds with torus boundaries.
problem Computing torsion function for 3-manifolds with specific boundary conditions.
method Defined adjoint torsion function on moduli stack of G-local systems, proved regularity condition, provided formula for product of PGL2-torsions.
result Computed adjoint PGSp4-torsions of figure-eight knot complement for boundary-unipotent local systems.
Extends Seifert algorithm to 3-manifolds via surgery.
problem Construct Seifert surfaces in arbitrary 3-manifolds.
method Uses surgery on framed links in S^3 to extend classical algorithm.
result Explicit construction of Seifert surfaces in 3-manifolds.
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
Infinite rank groups found in 3-manifolds with infinite fundamental groups.
problem Understanding the structure of diffeomorphism and homeomorphism groups of 3-manifolds with infinite fundamental groups.
method Analyzing actions of barbell diffeomorphisms on spaces of embedded arcs and configuration spaces.
result Groups of diffeomorphisms and homeomorphisms have infinite rank.
Character scheme of small Seifert 3-manifolds is reduced if and only if no exceptional abelian character exists.
problem Character scheme of small Seifert 3-manifolds and its reduction condition.
method Complete description of the SL2(C)-character scheme of small Seifert 3-manifolds.
result Character scheme is reduced if and only if no exceptional abelian character exists, and exceptional abelian character has multiplicity 2.
Complexity of counting group homomorphisms depends on group structure and presentation.
problem Computing the number of homomorphisms from a group to another.
method Obstruction theory and polynomial time algorithms for specific cases.
result Efficient algorithms for counting homomorphisms under certain conditions.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. Study skein modules via gauge theory, finding non-TQFT dimensions.
problem Understanding skein modules of 3-manifolds using gauge theory.
method Embed skein modules into 4d N=4 super-Yang-Mills theories, using N=1 supersymmetry. result Find non-standard dimensions of skein modules, differing from TQFT predictions.
The paper proves a combinatorial Ricci flow converges to hyperbolic structures on certain 3-manifolds.
problem Proving convergence of combinatorial Ricci flow to hyperbolic structures.
method Combinatorial Ricci flow on closed pseudo 3-manifolds with specific edge valences.
result Existence and uniqueness of a complete hyperbolic metric with totally geodesic boundary.
New topological restrictions found for spaces with nonnegative Ricci curvature.
problem Understanding topological properties of spaces with nonnegative Ricci curvature.
method Analyzing complete Riemannian manifolds and RCD(0,n) spaces, applying rigidity and vanishing theorems.
result Proved a Betti number rigidity theorem and a vanishing theorem for simplicial volume.
The study distinguishes exotic R^4's using knot Floer homology and Casson handles.
problem Distinguishing exotic R^4's with different knot Floer homologies.
method Using Gadgil's end Floer homology and Casson handles.
result A countably infinite family of pairwise nondiffeomorphic chiral exotic R^4's.
New group theory insights on knot surgery results.
problem Understanding non-simply connected 3-manifolds from Dehn surgery.
method Group theoretic analysis of Property P conjecture variations.
result New group theoretic perspectives on Dehn filling.
The paper proves a relation between four types of invariants.
problem Proving a precise relation between four types of invariants.
method Analyzing pseudo-Anosov homeomorphisms and cusped hyperbolic 3-manifolds at roots of unity.
result A precise relation between the Baseilhac-Benedetti invariants and the Bonahon-Liu-Wong-Yang invariants.