Knots connected via a trivial band sum to connected sum.
problem Conditions for band-connected sum to equal connected sum.
method Analyzing knots and bands to determine conditions for equality.
result A band is trivial if and only if a band-connected sum equals a connected sum.
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
Knots can only be fibered if each component is fibered and their genus is at least the connected sum.
problem Understanding the fibered nature of band-connected sums of knots.
method Analyzing the genus of fibered knots and their connected sums.
result The genus of a fibered knot is greater than or equal to the genus of its connected sum components.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
Connected sums of quasipositive links are quasipositive.
problem Understanding when connected sums of links are quasipositive.
method Filling disk technique
result Connected sums of quasipositive links are quasipositive.
Study knot Floer homology for connected sums, proving a formula and constructing a homomorphism.
problem Understanding knot Floer homology for connected sums of knots.
method Proved a formula for the conjugation action on knot Floer complexes of connected sums, constructed a homomorphism from smooth concordance group.
result Computed involutive invariants on large surgeries of connected sums of knots using the connected sum formula.
Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.
problem Behavior of Morse-Novikov number under knot operations.
method Additivity under connected sum and invariance under cabling.
result Morse-Novikov number is additive under connected sum and unchanged by cabling.
Defines a new method for combining curves and describes how certain properties behave.
problem Combining curves and understanding their properties.
method Generalized connected sum for plane curves.
result Describes behavior of Arnold invariants under the generalized connected sum.
Proofs knot homology connected sums using grid complexes.
problem Proving Künneth formula for knot Floer homology of connected sums.
method Constructs a quasi-isomorphism of grid chain complexes.
result Functorial behavior of Legendrian and transverse invariants under connected sum.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
New method proves Jones Polynomial's connect sum property.
problem Jones Polynomial's behavior under connect sums.
method Trip matrix method for calculating Jones Polynomial.
result Jones Polynomial is multiplicative under connect sums.
Invariants for connected sums of 3-manifolds, proving Furuta's theorem.
problem Computing and comparing Manolescu invariants of connected sums.
method Using inequalities and monopole theory to compute invariants.
result Proof of Furuta's Theorem using Manolescu invariants.
Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Formula proved for Lusternik-Schnirelmann category of connected sums of manifolds.
problem Calculating the Lusternik-Schnirelmann category for connected sums of manifolds.
method Used the Berstein-Hilton invariant to prove the formula.
result Proved the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for closed manifolds.
Study connects knot polynomials with number theory sums.
problem Alexander polynomials and Dedekind sums of torus knots.
method No specific method mentioned; connects known concepts.
result Established relationship between knot theory and number theory.
When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo iπ2 and the twi…
Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.
problem Determining the structure of Kauffman bracket skein module of connected sums of solid tori.
method Used algebraic methods over the ring of Laurent polynomials to prove the conjecture.
result Proved a conjecture about the Kauffman bracket skein module of connected sums of genus one handlebodies.
Paper constructs positive Ricci metrics on various connected sums of projective spaces.
problem No universal bound on Betti numbers for manifolds with positive Ricci curvature.
method Revisits and extends Perelman's techniques to construct metrics.
result Positive Ricci metrics constructed on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces.
Connected sum of CR manifolds with positive CR Yamabe constant is possible.
problem Establishing the existence of a CR structure with positive CR Yamabe constant for connected sums of CR manifolds.
method Analyzing the properties of connected sums of CR manifolds with positive CR Yamabe constant.
result The connected sum of M1 and M2 admits a CR structure with positive CR Yamabe constant. Connected sum of CR manifolds preserves positive CR Yamabe constant.
problem Understanding CR structures on connected sums of spherical CR manifolds.
method Analyzing spherical CR manifolds with positive CR Yamabe constant.
result Connected sum of CR manifolds with positive CR Yamabe constant also admits a spherical CR structure with positive CR Yamabe constant.
Weyl energy decreases for connected sums of certain four-manifolds.
problem Finding metrics with minimized Weyl energy on connected sums of four-manifolds.
method Proving existence of a metric on the connected sum with strictly smaller Weyl energy than the sum of energies of the original manifolds.
result Weyl energy of the connected sum is strictly smaller than the sum of energies of the original manifolds.
Formula connects knot complements' invariants.
problem Understanding invariants of knot complements.
method Proposed a connect sum formula for two-variable series invariants.
result Numerical evidence supports the formula for various torus knots.
We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…
4-manifolds can be exotic after connected sum with S^2 x S^2.
problem Existence of exotic contractible 4-manifolds.
method Constructed a cork that remains exotic after connected sum with S^2 x S^2.
result Existence of exotic pair of contractible 4-manifolds.
Classifies 4-manifolds up to connected sum with complex projective planes.
problem Classifying 4-manifolds up to connected sum with complex projective planes.
method Classification based on fundamental group, orientation character, and extension class involving second homotopy group.
result Closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of fundamental group, orientation character, and an extension class involving the second homotopy group.
Paper disproves a theorem about Kauffman bracket skein module structure.
problem Disproving a 22-year-old theorem about Kauffman bracket skein module structure.
method Analyzing handle slidings on compressing discs in handlebodies.
result More relations found than previously predicted for connected sum of handlebodies.
Study on LS-category and topological complexity of connected sum manifolds.
problem Behavior of LS-category and topological complexity under connected sum operation.
method Analyzes the invariants LS-category and topological complexity for connected sum of manifolds.
result For orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$; for simply connected manifolds, $\TC (M\# N)\ge\max\{\TC M,\TC N\}$.
Knots formed from torus knots are not concordant to L-space knots.
problem Understanding concordance in knots formed from connected sums of torus knots.
method Analyzing properties of knots formed from connected sums of torus knots.
result Knots formed from torus knots are not concordant to L-space knots.
Knot surgeries on 4-manifolds are shown to be diffeomorphic after connected sum with CP^2.
problem Characterizing diffeomorphism classes of knot surgeries in 4-manifolds.
method Proving diffeomorphism of knot surgeries after connected sum with CP^2.
result Knot surgeries on 4-manifolds are mutually diffeomorphic after a connected sum with CP^2.
Formula connects Heegaard Floer homology terms for 3-manifolds.
problem Formula for involutive Heegaard Floer homology terms.
method Connected sum formula for involutive Heegaard Floer homology.
result Example of 3-manifold with non-standard correction terms.
New theorem on positive scalar curvature for connected sums of manifolds.
problem Conditions for manifolds to admit metrics of positive scalar curvature.
method Index theory for Dirac operators
result Potential generalization of Gromov-Lawson's theorem to nonspin manifolds.
New G_2-manifolds created from quotients of twisted connected sums.
problem Creating new G_2-manifolds with specific topological properties.
method Generalizing Kovalev's twisted connected sums by taking quotients of pieces before gluing.
result Realization of a wider range of topological types of G_2-manifolds.
Study of Pin(2)-monopole Floer homology under connected sums.
problem Understanding Pin(2)-monopole Floer homology behavior under connected sums.
method Constructing a partially defined A-infinity module structure and using Eilenberg-Moore spectral sequences.
result Identify the Floer chain complex of a connected sum via an A-infinity tensor product of modules.
New G2-instanton found on complex bundle.
problem Constructing G2-instantons on complex bundles. method Using earlier work and a new bundle construction.
result Irreducible unobstructed G2-instanton found. Connected sum of manifolds preserves Ricci lower bounds.
problem Proving connected sum of manifolds with spectral Ricci lower bounds.
method Geometric construction resembling Gromov-Lawson tunnel, focusing on γ>n−2n−1. result Connected sum M#N also admits a metric satisfying the Ricci lower bound condition. We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.
The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.
problem Understanding the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
method Combinatorial operations and formulas for the number of terms in the connected sums of sphere products.
result Formulas for the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.
problem Defining and analyzing the adjoint Reidemeister torsion for connected sums of knots.
method Defined a natural way to compute the adjoint Reidemeister torsion for high-dimensional components of the character variety.
result The adjoint Reidemeister torsion is locally constant and satisfies the vanishing identity.
The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.
problem Understanding grid homology for spatial graphs with various types of edges.
method Developed grid homology for spatial graphs with cut edges and applied it to prove a Künneth formula for connected sums.
result A Künneth formula for knot Floer homology of connected sums is proven using grid homology.
The study allows for connected sums in manifolds with positive intermediate Ricci curvature.
problem Performing connected sums in manifolds with positive intermediate Ricci curvature.
method Introducing and utilizing k-core metrics to show the possibility of connected sums. result Connected sums are possible under certain conditions involving k-core metrics. The Strong Slope Conjecture is proven for specific knot types.
problem Proving the Strong Slope Conjecture for various knot types.
method Using connect sums and cabling, the conjecture is shown to be closed under these operations.
result The Strong Slope Conjecture is established for graph knots.
Study on geodesics on connected sums of manifolds, showing infinite number of distinct closed geodesics.
problem Counting geometrically distinct closed geodesics on connected sums of manifolds.
method Analyzing the asymptotics of the number of closed geodesics on Riemannian or Finsler metrics.
result Generic Riemannian metrics on compact 3-manifolds have infinitely many geometrically distinct closed geodesics.
The paper improves bounds on topological complexity for certain manifolds.
problem Determining bounds on topological complexity for specific manifolds.
method Analyzing cohomology classes and their pullbacks, using Gromov norm.
result Improved bounds on topological complexity for connected sums.
New Spin(7)-manifolds constructed via generalized connected sums.
problem Creating compact Spin(7)-manifolds for geometric engineering.
method Generalized connected sum of Calabi-Yau four-fold and G2-holonomy manifold.
result Construction of new compact Spin(7)-manifolds with holonomy.
New homology theory for instantons, defined and analyzed.
problem Understanding instanton structures in symplectic geometry.
method Definition and analysis of twisted Symplectic Instanton homology, fitting into Floer Field theory.
result Properties of the new homology invariant for connected sums and Dehn surgery.
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …