The study examines obstructions to links being shake slice.
problem Understanding when links are not shake slice.
method Examined shake concordance and zero surgery manifolds, and provided obstructions based on Arf invariants and algebraic sliceness.
result Links that are shake concordant have homology cobordant zero surgery manifolds, and provided specific obstructions to shake sliceness.
Regular sliceness implies once-stably decomposable sliceness in symplectizations.
problem Relationship between regular and decomposable Lagrangian cobordisms in symplectizations.
method Stabilization-free strategy and satellite operations.
result Regular sliceness implies once-stably decomposable sliceness.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge `horocyclically' to 2 then associated linear slices converge…
We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results.
New bounds on knot complexity defined via band number and surface diagrams.
problem Understanding the complexity of knots and their generalizations.
method Defining and analyzing band number, using broken surface diagrams and 3-manifold embeddings.
result Band number is an upper bound on double slice genus, a knot complexity measure.
Condition for generic sliced spaces to be globally hyperbolic.
problem Global hyperbolicity of sliced spaces without lapse or shift function hypotheses.
method Topological condition for globally hyperbolic sliced spaces.
result Generic sliced spaces are globally hyperbolic under certain conditions.
New PAC-Bayesian bounds improve Sliced-Wasserstein distances.
problem Improving statistical properties of Sliced-Wasserstein distances.
method Leveraging PAC-Bayesian theory to provide bounds and learning procedures.
result PAC-Bayesian generalization bounds for adaptive SW distances.
New examples show algebraically slice knots with specific genus bounds.
problem Understanding slice genus of algebraic knots and their mirrors.
method Genus bound from Casson-Gordon invariants and cabling formula.
result Examples of algebraically slice knots with specific genus bounds.
The study classifies χ−slice pretzel links and Seifert fiber spaces.
problem Understanding χ−slice pretzel links and their properties. method Analyzing the sliceness of pretzel knots and extending results to pretzel links.
result Complete classifications of positive and negative pretzel links that are χ−slice, and partial classifications of 3-stranded and 4-stranded pretzel links. Defines concordance for spatial graphs and proves sliceness equivalence.
problem Understanding concordance and sliceness for spatial graphs.
method Smooth definitions and linking number conditions.
result Sliceness of a spatial graph is equivalent to a condition on linking numbers and a link.
Identifies half-space neighborhoods of pleating rays in the Riley slice of Schottky groups.
problem Determining points in the Riley slice of Schottky groups.
method Adapting ideas from L. Keen and C. Series, identifying half-space neighborhoods of pleating rays.
result Provides a provable method to determine if a point is in the Riley slice.
We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoo…
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.
The paper generalizes the T-genus to characterize slice knots and slice genus.
problem Characterizing slice knots and slice genus using the T-genus. method Generalizing the T-genus to provide a 3-dimensional characterization of the slice genus. result The difference between the T-genus and the slice genus can be arbitrarily large. A new method optimizes projection directions for sliced Wasserstein distances.
problem Finding informative projecting directions for sliced Wasserstein distances is computationally expensive.
method Amortized projection optimization to predict directions efficiently.
result Proposed amortized models improve generative modeling performance.
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice…
Extends SW and GSW to compare heterogeneous joint distributions.
problem Limited applicability of SW and GSW to heterogeneous joint distributions.
method Introduces HHRT and PGRT to extend SW and GSW.
result H2SW distance for heterogeneous joint distributions.
The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely generated free (abelian) subgroup. Here it is shown that the concordance group of algebr…
A new method for comparing image probability measures using convolution operators.
problem Efficiently comparing images using conventional sliced Wasserstein methods.
method Proposed convolution sliced Wasserstein (CSW) methods with stride, dilation, and non-linear activation.
result CSW demonstrates favorable performance over conventional sliced Wasserstein in image comparison and deep generative modeling.
Study on r−shake slice knots and proves 0-shake slice knots are slice.
problem Understanding and characterizing r−shake slice knots. method Exploring the relation to corks and proving slice properties.
result Proves 0-shake slice knots are slice.
A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-ma…
Proves certain knots are slice without shaking.
problem Identifying slice knots without using traditional methods.
method Direct proof for 0−shake slice knots. result Proves 0−shake slice knots are slice. CB-SLICE identifies concept-based error slices in deep learning models.
problem Systematic errors in deep learning models on specific groups.
method Concept Bottleneck Models (CBMs) and concept representations.
result CB-SLICE outperforms state-of-the-art methods in error slice identification.
New bounds improve neural network generalization through slicing.
problem Difficulty in evaluating mutual information in high dimensions for neural networks.
method Slicing the parameter space and using disintegrated mutual information and k-sliced mutual information.
result Slicing improves generalization and offers significant computational and statistical advantages.
We define an operation on homology B4 which we call an n-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples we…
A new distance measure balances projection exploration and informativeness.
problem Inefficient and incomplete projection sampling in existing sliced-Wasserstein distances.
method Proposes Distributional Sliced-Wasserstein (DSW) that optimally balances projection exploration and informativeness.
result DSW generalizes Max-SW and can be computed efficiently.
Proves a special knot type is slice.
problem Characterizing slice knots.
method Proof by contradiction and algebraic topology.
result 0-shake slice knots are indeed slice.
The paper explores statistical and topological properties of sliced probability divergences.
problem Understanding the topological, statistical, and computational consequences of slicing divergences.
method Deriving theoretical properties of sliced probability divergences, including metric axioms preservation and weak continuity.
result Sliced divergences share similar topological properties and have stable sample complexity.
The Conway knot is not slice, resolving a knot classification problem.
problem Determining which knots are slice in 4-dimensional space.
method Demonstrated through a proof involving knot classification and properties of slice knots.
result The Conway knot is the first example of a non-slice knot that is topologically slice and a positive mutant of a slice knot.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
New bounds on slice genus from knot invariants.
problem Bounding slice genus of knots in RP3. method Using s-invariant to establish lower bounds. result Proves conjecture on slice genus bounds.
Paper discusses sliced generative models for improved sample discrimination.
problem Improving sample discrimination in generative models.
method One-dimensional slicing of AutoEncoder-based generative models, focusing on normality tests and classical distances.
result The second group of methods based on classical distances gives a faster decrease rate of Fréchet Inception Distance (FID).
Study shows knots can be reversed without becoming trivially concordant.
problem Understanding when knots can be reversed without changing their concordance class.
method Constructed an infinite family of knots, analyzed their concordance properties.
result Found an infinitely generated free subgroup in the concordance group of topologically slice knots.
The paper discusses when spacetimes have constant mean curvature slices, an important but unknown condition.
problem The existence of constant mean curvature (CMC) Cauchy surfaces in spacetimes is crucial but not always guaranteed.
method Expository review of existing results and conjectures about CMC slices in spacetimes.
result It is not known whether spacetimes with CMC slices are generic, highlighting an important open problem.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.
Formula connects surface and curve invariants via slice transitions.
problem Computing surface invariants from curve invariants.
method Introducing differential measures for local changes across singular slice transitions.
result Explicit formula for surface invariant change during quadruple-point events.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
The study explores deep and shallow slice knots in 4-manifolds, linking them to conjectures and proving existence and nonexistence results.
problem Understanding slice knots in 4-manifolds and their properties.
method Using Wall self-intersection invariant and Rohlin's result, the study examines various 4-manifolds and their boundaries to find deep slice knots and prove nonexistence results.
result Every 4-manifold with one 0-handle and any number of 2-handles has a deep slice knot in its boundary.
Proposes a new distance metric for multi-marginal optimal transport.
problem Computational scalability in multi-marginal optimal transport.
method Random one-dimensional projections to construct sliced multi-marginal Wasserstein distance.
result Sliced multi-marginal Wasserstein distance is a metric with dimension-free sample complexity.
The paper studies CMC foliations and their conformal aspects on Riemannian manifolds.
problem Understanding and normalizing CMC foliations on conformally compact manifolds.
method Non-linear PDEs and conformal transformations.
result Locally, any slicing can be made into a CMC foliation by conformal changes.
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
New method freely slices good boundary links with specific conditions.
problem Slicing good boundary links with multiple components.
method Using a Seifert surface and homotopically trivial plus assumption.
result Provides new freely slice links and subsumes previous methods.
We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n>1 the (n+1)-st iterated Bing double of a knot is rationally slice if and only if the n-th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying …
The paper defines and studies new types of links in 4-manifolds.
problem Understanding and classifying links in 4-dimensional spaces.
method Defining and studying r-shake slice and r-shake concordant links, providing examples and characterizations.
result Characterizations of shake slice and shake concordant links for r=0, and proving invariants of shake concordance.
Characterizes values of slice-torus invariants related to knot genus.
problem Understanding the values of slice-torus invariants for knots.
method Characterization based on stable smooth slice genus.
result Existence of slice torus invariants without explicit constructions.