Study of knots in 4-manifolds, proving slice genus is not always an invariant.
problem Slice genus is not always an invariant of X0(K). method Analyzing 0-shake genus and using satellite operations. result Infinitely many knots with 0-shake genus strictly less than slice genus. Characterizes the Z-genus of boundary links using Blanchfield forms.
problem Determining the Z-genus of boundary links.
method Characterization using Blanchfield forms.
result Variant of shake genus equals Z-genus of knot.
We give a method for obtaining infinitely many framed knots which represent a diffeomorphic 4-manifold. We also study a relationship between the n-shake genus and the 4-ball genus of a knot. Furthermore we give a construction of homotopy 4-spheres from a slice knot with unknotting number one.
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.
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.
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.
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…
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. SHAKE-GNN scales GNNs for large graphs with multi-scale representations.
problem Scaling Graph Neural Networks (GNNs) to large graphs.
method SHAKE-GNN uses a hierarchy of Kirchhoff Forests for stochastic multi-resolution graph decompositions.
result SHAKE-GNN achieves competitive performance on large-scale graph classification benchmarks.
New complexity measure for shake-slice knots established.
problem Defining and measuring complexity for shake-slice knots.
method Using dualizable patterns and studying knot signatures.
result Existence of n-shake-slice knots with specified complexity. 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.
Let LHT be a left handed trefoil knot and K be any knot. We define Mn(K) to be the homology 3-sphere which is represented by a simple link of LHT and LHT♯K with framings 0 and n respectively. Starting with this link, we construct homotopy K3 and spin rational homology K3 surfaces containing …
A new RL framework handles autocorrelated actions for better learning and stability.
problem Improving reinforcement learning algorithms for better stability and efficiency.
method Introduces a new algorithm that optimizes policies with autocorrelated actions.
result The new algorithm outperforms existing methods in four simulated control problems.
Averaging SGD weights improves deep learning models.
problem Improving deep learning model generalization.
method Simple averaging of SGD weights with cyclical or constant learning rate.
result SWA leads to better generalization and flatter solutions.
Khovanov homology distinguishes exotic 4-manifolds.
problem Existence of exotic compact orientable 4-manifolds.
method Khovanov-Rozansky gl2 skein lasagna module and related invariants. result First analysis-free proof of exotic 4-manifolds.
Proposes NCMN to remove feature correlation in multiplicative noise.
problem High feature correlation in multiplicative noise reduces network performance.
method Exploits batch normalization to remove correlation effect.
result Significantly improves performance on image classification tasks.
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis…
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
The paper characterizes when a 2-sphere can be embedded in a knot trace.
problem Characterizing when a 2-sphere can be embedded in a knot trace.
method Using classical and computable knot invariants, the paper provides conditions for embedding a 2-sphere in a knot trace.
result Conditions for a knot to be topologically n-shake slice. There is intense interest in understanding the stochastic and dynamical properties of the global Foreign Exchange (FX) market, whose daily transactions exceed one trillion US dollars. This is a formidable task since the FX market is characterized by a web of fluctuating exchange rates, with subtle inter-dependencies wh…
Paper introduces TAP-Vid, a benchmark for tracking any point in videos.
problem Tackles the problem of tracking arbitrary physical points on surfaces over longer video clips.
method Formalizes the problem as TAP, introduces TAP-Vid benchmark, uses crowdsourced pipeline with optical flow estimates, proposes TAP-Net model.
result TAP-Net outperforms all prior methods on TAP-Vid benchmark when trained on synthetic data.
Robot learns to imitate human interactions through deep learning.
problem Teaching robots to coordinate actions with human partners.
method Deep learning framework for motion embedding, prediction, and trajectory generation.
result Importance of predictive and adaptive components for successful imitation.
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
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. Paper bounds double slice genus of knots.
problem Understanding knot genus complexities.
method Using Casson-Gordon invariants, the paper defines and bounds the double slice genus.
result Double slice genus can be much larger than slice genus.
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
Study shows four-genus ratio of two-bridge knots decreases as knots get more complex.
problem Understanding the relationship between smooth four-genus and Seifert genus in two-bridge knots.
method Analytical proof focusing on two-bridge knots and their crossing numbers.
result The expected value of the ratio between smooth four-genus and Seifert genus tends to zero as the crossing number increases.
New examples of local signatures for genus 2 and 3 fibrations.
problem Finding unusual local signatures for specific fibrations.
method Providing specific examples of local signatures for genus 2 and 3 fibrations.
result Examples of local signatures different from usual ones for genus 2 and 3 fibrations.
Classifies 3-braid knots with maximal 4-genus using McCoy's method.
problem Classifying 3-braid knots with maximal 4-genus.
method McCoy's twisting method and Xu normal form.
result Upper bounds for the topological 4-genus of 3-braid knots.
New link invariant bounds topological slice genus.
problem Bounding the topological slice genus of links.
method Introducing algebraic genus as a new link invariant and using Casson-Gordon invariants.
result Algebraic genus is an upper bound for the topological slice genus.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
problem Analyzing Khovanov homology of Turaev genus one links.
method Examined Khovanov homology in specific polynomial gradings.
result Found a trivial summand in Khovanov homology of Turaev genus one links.
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.
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…
This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve grap…
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
Classifies Teichmüller curves in genus three from genus two.
problem Classifying Teichmüller curves in genus three from genus two.
method Action on homology modulo two of affine groups of translation surfaces.
result Classification of orbits of square-tiled surfaces.
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. Positive braid knots with maximal signature have maximal 4-genus.
problem Determining knots with maximal 4-genus.
method Analyzing positive braid knots and their signature invariants.
result All positive braid knots with up to 12 crossings have maximal 4-genus.
Survey of minimal genus problem progress.
problem Finding the smallest genus surface in 3-manifolds.
method Survey and review of existing research.
result Summarizes recent advancements in the minimal genus problem.
Algorithm calculates genus of embedded graphs on surfaces.
problem Finding minimal and maximal genus for graph embeddings.
method Constructing special branched coverings of the 2-sphere.
result Algorithm calculates orientable genus and maximal genus of graphs.
Classifies changes between genus one fibered knots.
problem Understanding changes between genus one fibered knots.
method Analyzed all monodromies and determined manifolds.
result Classified all generalized crossing changes between genus one fibered knots.