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.
Just as war is sometimes fallaciously represented as a zero sum game -- when in fact war is a negative sum game - stock market trading, a positive sum game over time, is often erroneously represented as a zero sum game. This is called the "zero sum fallacy" -- the erroneous belief that one trader in a stock market exch…
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.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…
Defines a universal state sum construction for various TQFTs.
problem No specific problem stated; universal construction for TQFTs.
method Defines a universal state sum construction using n-categories with specific conditions.
result Produces state sums from n-categories and handle decompositions of n+1-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.
Study connects geodesic sums to area constant.
problem Understanding geodesic sums in Teichmüller space.
method Relates trimmed sums of twists to area Siegel-Veech constant.
result Established connection between geodesic sums and area constant.
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.
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…
Deep sum-product networks learn faster than shallow models.
problem The speed of parameter optimization in sum-product networks.
method Theoretical analysis and empirical experiments on overparameterized sum-product networks.
result Gradient-based optimization in deep sum-product networks is equivalent to gradient ascent with adaptive and time-varying learning rates and additional momentum terms.
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.
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± and St behave under the generalized connected sums.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
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.
Summing Hamiltonian manifolds with a common submanifold.
problem Combining Hamiltonian manifolds with a shared submanifold.
method Establishing symplectic reduction and comparing Chern classes.
result Symplectic reduction of the sum agrees with the sum of reductions.
We prove for any positive integer n there exist boundary-sum irreducible Zn-corks with Stein structure. Here `boundary-sum irreducible' means the manifold is indecomposable with respect to boundary-sum. We also verify that some of the finite order corks admit hyperbolic boundary by HIKMOT.
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.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non…
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.
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 methods optimize sums of bivariate functions on finite domains.
problem Optimizing functions with multiple arguments that are sums of bivariate functions.
method Measure-valued extensions, ℓ2-approximation, entropy-regularization, linear programming, coordinate ascent. result Tractable problem formulations solvable with various methods.
This work classifies belted sum decompositions of fully augmented links.
problem Understanding belted sum decompositions of fully augmented links.
method Explicit classifications of thrice punctured spheres in FAL complements, geometric, combinatorial, and diagrammatic characterizations.
result Every FAL complement canonically decomposes into FALs which are either prime or two-fold covers of the Whitehead link.
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.
The paper studies knot Floer homology under Murasugi sum and establishes graded isomorphisms.
problem Behavior of knot Floer homology under Murasugi sum.
method Established a graded version of Ni's isomorphism and proved τ=g for each summand.
result Graded isomorphisms between extremal knot Floer homologies of Murasugi sum and tensor products.
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is ag…
Study end sum for surfaces and prove uniqueness results.
problem Uniqueness of end sum for surfaces and related manifolds.
method Analyzing end sum and adding a 1-handle at infinity for surfaces.
result The end sum of two surfaces with compact boundary is uniquely determined by the chosen ends.
Derives exact formula for Minkowski sum of ellipsoids in N-space.
problem Finding volume bounds for Minkowski sum of ellipsoids.
method Closed-form parametric equation derivation and volume bounds calculation.
result Upper and lower volume bounds for Minkowski sum of ellipsoids.
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
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 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.
Summing over 3-manifolds using TQFT partition functions.
problem Summing over all 3-manifolds with fixed boundary.
method Rewriting the sum over 3-manifolds as a sum over homology groups, using TQFT partition functions and topological boundary conditions.
result Existence of a distribution of 2d TQFTs whose ensemble average equals the sum over 3-manifolds.
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.
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.
Researchers found multiple ways to end-sum 4-manifolds, contradicting a previous conjecture.
problem Nonuniqueness of end-sums in 4-manifolds.
method Explicit examples and detailed discussion of end-cohomology algebra.
result Uncountably many distinct proper homotopy types from end-sums.
Groupoids help define Riemann sums on manifolds.
problem Defining Riemann sums on compact manifolds.
method Using groupoids and the van Est map.
result Riemann sums converge to the usual integral.
Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other …
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. Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …
Sum-of-norms clustering recovers mixtures of Gaussians even with infinite samples.
problem Recovering a mixture of Gaussians from a large number of samples.
method Sum-of-norms clustering with equal weights, convex optimization.
result Sum-of-norms clustering can recover mixtures of Gaussians even as the number of samples tends to infinity.
Knots can be constructed and decomposed using Murasugi sums of Seifert surfaces.
problem Understanding the structure of knots through Murasugi sums.
method Using Murasugi sums to decompose and construct knots, showing the structure of a bi-directed complete graph.
result Any knot can be a Murasugi sum of any two knots, with bounds on minimal complexity.
In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic 4-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from o…
We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
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…
State sums for quantum link invariants from a specific representation.
problem Calculating quantum link invariants from a specific representation of U_q(gl_{N|M}).
method Using state sums and representation theory of U_q(gl_{N|M}).
result Explicit relation with Kashaev invariants for the N-th exterior power of the standard representation.
Nahm sums are q-series of a special hypergeometric type that appear in character formulas in Conformal Field Theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in Quantum Knot Theory, namely we prove the stability of th…
Jones polynomials compute weighted sums of Lefschetz numbers.
problem Computing Lefschetz numbers for braids.
method Colored Jones polynomials of braid closures.
result Jones polynomials compute abelianized Lefschetz numbers.
Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.