arXiv research

Connected sum affects crossing numbers of flat virtual knots.

Classifies exceptional Legendrian realizations of Hopf link connected sums.

We show that a band-connected sum of knots $K_0$ and $K_1$ along a band $b$ is equal to the connected sum $K_0\# K_1$ if and only if $b$ is a trivial band.

Proofs knot homology connected sums using grid complexes.

New method proves Jones Polynomial's connect sum property.

Genus 3 Heegaard groups of lens space connected sums are finitely generated.

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Study connects knot polynomials with number theory sums.

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 $iπ^2$ and the twi…

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

Weyl energy decreases for connected sums of certain four-manifolds.

Formula connects knot complements' invariants.

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.

Paper disproves a theorem about Kauffman bracket skein module structure.

We give inequalities for the Manolescu invariants $α,β,γ$ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a $\mathbb{Z}^\infty$ subgroup of the homology cobordism …

We show that if a fibered knot $K$ is expressed as a band--connected sum of $K_1, \ldots, K_n$, then each $K_i$ is fibered, and the genus of $K$ is greater than or equal to that of the connected sum of $K_1,\ldots,K_n$.

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.

Connected sum of manifolds preserves Ricci lower bounds.

The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.

The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.

The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.

The study allows for connected sums in manifolds with positive intermediate Ricci curvature.

The paper improves bounds on topological complexity for certain manifolds.

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 …

We investigate the boundness of the Riesz transform on $L^p$ for connected sum of manifolds where the Riesz transform is bounded on $L^p$.

The study preserves positive Ricci curvature on connected sums of fibre bundles.

The study characterizes 3D manifolds using specific Morse-Bott functions.

We show the Morse-Novikov number of knots in $S^3$ is additive under connected sum and unchanged by cabling.

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

We prove that the connected sum of two links is quasipositive if and onlyif each summand is quasipositive. The prove is based on the filling disk technique

Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on …

If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.

In this paper, we provide the necessary and sufficient conditions for the connected sum of knots in $S^3$ to be Legendrian simple.

Study on knot unknotting numbers and their behavior under connected sums.

We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.

New 5-manifolds allow Sasaki-Einstein structures.

The paper studies how knots and links behave under connected sum operations.

We show that the m-fold connected sum $m\#\mathbb{C}\mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.

In this paper, we shall prove that any Heegaard splitting of a $\partial$-reducible 3-manifold $M$, say $M=W\cup V$, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of $n$ manifolds $M_{1},..., M_{n}$ where $M_{i}$ is either a solid torus or a $…

Connected sums defined for codimension two locally flat submanifolds in higher dimensions.

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homolo…

Stability of Yang-Mills connections' Morse indices and nullity in 4D.

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of t…

Proves a formula for a special invariant of 4-manifolds.