Bound on splitting number in terms of four-genus of knots.
problem Understanding the relationship between splitting number and four-genus of knots.
method Provided a bound on the splitting number in terms of the four-genus of related knots.
result Bound on splitting number in terms of four-genus.
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
Lower bounds on splitting number derived from signature and nullity.
problem Determining the minimal number of crossing changes to split a link.
method Using multivariable signature and nullity to derive lower bounds.
result Efficiently computed splitting numbers for many links.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
Paper generalizes splitting number to classify plane curve arrangements.
problem Distinguishing embedded topology of plane curves.
method Introduces splitting graph to generalize splitting number.
result Classifies embedded topology of specific plane curve arrangements.
In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s-1, and less than or equa…
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
Algorithm determines unlinking number 1 for certain links.
problem Determining unlinking number 1 for links.
method Algorithm to find crossing changes turning links into split links.
result Algorithm determines unlinking number 1 for certain links.
The paper establishes conditions for link invariants to bound the weak splitting number.
problem Determining the minimum number of crossing changes to split a link into knots.
method Conditions on R-valued link invariants, link Floer homology, and known obstructions. result New bounds on the weak splitting number for many prime links.
New formula splits instanton Floer homology invariants.
problem Expressing Seiberg-Witten invariants in terms of other invariants.
method Observation of a splitting formula in instanton Floer homology.
result A new formula for instanton Floer homology invariants.
Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving …
New link concordance invariants from perturbed sl(n) homology give lower bounds on slice genus and splitting number.
problem Link concordance and slice genus determination.
method Defining a family of invariants sn from perturbed sl(n) homology. result Computes slice genus of positive links and determines splitting number of positive torus links.
This paper compares communication efficiency of split learning and federated learning in various scenarios.
problem Comparing communication efficiency of split learning and federated learning in different settings.
method Examined various practical scenarios of distributed learning setups and compared the two methods.
result Communication efficiency of split learning and federated learning depends on the number of clients, model size, and data samples.
Study shows that splitting links requires an arbitrarily large number of extra crossings.
problem The problem is to determine the minimum number of extra crossings needed to transform a diagram of a split link into a split diagram.
method The approach uses Reidemeister moves and the framework of bubble tangles, along with techniques from Riemannian geometry.
result There exist split links with diagrams requiring an arbitrarily large number of extra crossings.
The study uses Heegaard Floer homology to analyze and bound the splitting number of links.
problem Bounding the splitting number of links using Heegaard Floer homology.
method Constructing a cobordism and applying d-invariant inequalities to link Floer homology. result A Heegaard Floer theoretical criterion for bounding the splitting number of links, especially effective for L-space links.
We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken 3-manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer n there…
Proposes SCD-split for CP to balance interpretability and efficiency.
problem Difficult interpretation of disconnected subintervals in CP prediction sets.
method Incorporates smoothing operations into CP framework.
result SCD-split balances interval length and subinterval number, theoretically provable.
This paper tackles target-dependent label complexity gap in active learning.
problem Target-dependent label complexity gap in Agnostic Active Learning.
method Introduces a novel distribution-splitting strategy based on number density to reduce label complexity and error rate.
result Provides theoretical guarantees and practical advantages for reducing label complexity and error rate.
Subagging improves regression tree performance, especially with many splits.
problem Improving regression tree performance with subsample aggregating.
method Formalized bias and variance dependencies, compared subagging to single trees, and analyzed optimal tree sizes.
result Subagging improves tree performance, especially with many splits.
Derives ideal train/test split for ridge regression in large data limit.
problem Finding optimal train/test split for ridge regression in large data scenarios.
method Mathematical derivation of optimal train/test split, considering ridge tuning parameter and asymptotic behavior.
result The optimal train/test split for ridge regression in the large data limit depends weakly on the ridge tuning parameter alpha.
Let T be a separating incompressible torus in a 3-manifold M. Assuming that a genus g Heegaard splitting V∪SW can be positioned nicely with respect to T (e.g. V∪SW is strongly irreducible), we obtain an upper bound on the number of stabilizations required for V∪SW to become isotopic to a…
Identifies all 2-bridge links up to 11 crossings.
problem Classifying 2-bridge links up to a certain number of crossings.
method Enumerated all 2-bridge links up to 11 crossings and calculated splitting numbers.
result Identified all 2-bridge links up to 11 crossings and calculated splitting numbers.
Law of large numbers for volumes of random 3-manifolds.
problem Understanding volumes of random 3-manifolds.
method Law of large numbers for random hyperbolic mapping tori and Heegaard splittings.
result Sharp answer to Dunfield and Thurston's conjecture.
Prime homology detects split links in prime characteristic.
problem Detecting split links in prime characteristic.
method Uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients.
result Reduced sl(P) link homology detects split links in Z/P. Geometric structures over algebras describe geodesics and spaces.
problem Understanding geodesics in hyperbolic and Euclidean spaces over non-standard algebras.
method Study geometric structures from Hermitian forms on real algebras (dual numbers, split-complex, split-quaternions).
result Presented a projective model for the hyperbolic bidisc.
The paper studies Betti numbers of HOMFLYPT homology modules for links.
problem Understanding the structure of HOMFLYPT homology modules and their Betti numbers.
method Analyzing the module structure and Betti numbers of the middle HOMFLYPT homology for links.
result The Betti numbers of the middle HOMFLYPT homology modules are finite and can be used to recover the Poincaré polynomial.
In this article, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two $(1,1)…
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
Optimal data split ratio is sqrt(p):1 for linear regression.
problem Lack of clear guidance on optimal training/testing data split ratio.
method Showed that optimal ratio is sqrt(p):1 for linear regression.
result Optimal ratio for training/testing split is sqrt(p):1.
On realizations of the complex Lie groups (E6,R)C,(E6,C)C,(E6,H)C and those real formsmath.RA The paper explores different realizations of complex Lie groups using various number fields.
problem Defining and understanding Lie groups with different number fields.
method Using Cayley algebras and fields of real numbers, complex numbers, split complex numbers, quaternions, and split quaternions to define and study Lie groups.
result The structure of Lie groups (E6,R)C,(E6,C)C,(E6,H)C and their real forms are determined. The paper explores actions of surface mapping class groups on 3-manifolds.
problem Understanding when the natural surjection from homeomorphisms to mapping class groups splits.
method Analyzing circle bundles and their properties over surfaces.
result The homomorphism does not split in many cases where the Euler characteristic divides the Euler number.
Memory split advantage: thinner networks outperform a single wide network.
problem Optimizing deep learning models with limited memory.
method Investigated training a single wide network vs. an ensemble of thinner networks with the same total number of parameters.
result An ensemble of several thinner networks outperforms a single wide network for large memory budgets.
The paper studies Heegaard splittings of knots with high bridge distance and finds stabilization patterns.
problem Stabilization patterns of Heegaard splittings for knots with high bridge distance.
method Analysis of minimal genus Heegaard splittings and their equivalence under stabilizations.
result Two splittings from the same family become equivalent after at most one stabilization, and splittings from different families require up to n−1 stabilizations. One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is rep…
Let K1,K2 be two knots with t(K1)+t(K2)>2 and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a comp…
We give an upper bound for the Reidemeister-Singer distance between two Heegaard splittings in terms of the genera and the number of cusp points of the product map of Morse functions for the splittings. It suggests that a certain development in singularity theory may lead to the best possible bound for the Reidemeister…
Invariant counts maximum stable umbilic splits.
problem Counting stable umbilic splits on surfaces.
method Introducing an invariant to count maximum stable umbilic splits.
result Established properties of the multiplicity invariant.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …
We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…
We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-…
Integrally splits L-spectra of integers into simpler components.
problem Understanding the homotopy type of L-spectra of integers.
method Using Anderson duality and splitting into simpler spectra.
result Splits L-spectra of integers into simpler components.
Classifies involutions on alternating prime non-split links.
problem Classifying involutions on alternating prime non-split links.
method Using the category of flypes between reduced alternating diagrams.
result Quotient of an alternating periodic link is alternating; freely 2-periodic alternating links have an even number of components.
The study identifies GOF-knots in 3-manifolds with reducible genus two Heegaard splittings.
problem Determining GOF-knots in 3-manifolds with reducible genus two Heegaard splittings.
method Using a necessary and sufficient condition for decomposing a genus two handlebody, the study identifies GOF-knots by analyzing simple closed curves on the Heegaard surface.
result The number and positions of GOF-knots in 3-manifolds with reducible genus two Heegaard splittings are determined.
This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. T…
Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two pa…
Given a link in S3 we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restric…