arXiv research

Study on $r-$shake slice knots and proves 0-shake slice knots are slice.

The study examines obstructions to links being shake slice.

Proves certain knots are slice without shaking.

We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a…

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 a special knot type is slice.

New complexity measure for shake-slice knots established.

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-h…

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.

The paper characterizes when a 2-sphere can be embedded in a knot trace.

SHAKE-GNN scales GNNs for large graphs with multi-scale representations.

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

Khovanov homology distinguishes exotic 4-manifolds.

Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link, we construct homotopy $K3$ and spin rational homology $K3$ surfaces containing …

Characterizes the Z-genus of boundary links using Blanchfield forms.

Deep neural networks are typically trained by optimizing a loss function with an SGD variant, in conjunction with a decaying learning rate, until convergence. We show that simple averaging of multiple points along the trajectory of SGD, with a cyclical or constant learning rate, leads to better generalization than conv…

A new RL framework handles autocorrelated actions for better learning and stability.

Multiplicative noise, including dropout, is widely used to regularize deep neural networks (DNNs), and is shown to be effective in a wide range of architectures and tasks. From an information perspective, we consider injecting multiplicative noise into a DNN as training the network to solve the task with noisy informat…

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…

New findings on knots that are both topologically and rationally slice.

Regular sliceness implies once-stably decomposable sliceness in symplectizations.

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…

The paper defines new knot genera and finds bounds for stabilization distances.

New knots found with tough, unsliceable discs.

A new slicing method speeds up sliced Wasserstein estimation.

A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice an…

Khovanov homology fails to differentiate certain slice disks.

Characterizes values of slice-torus invariants related to knot genus.

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. …

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

Study slice-regular polynomial functions via twistor space group actions.

The paper shows some Montesinos links can't be doubly sliced strongly.

Study shows most knots in a family are not slice.

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.

Study on slicing knots in 4-manifolds, focusing on CP^2-slicing numbers.

New invariant measures doubly slice links, disproving previous bounds.

New knots show linear independence in slice concordance.

Study shows certain knots can't be sliced using 2-fold branched covers.

Bing doubling is an operation which produces a 2-component boundary link B(K) from a knot K. If K is slice, then B(K) is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B(K) is boundary slice, then K is algebraically slice. We also show that the Ras…

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to …

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most…

Paper explores properties of slice-matching operators for measure transfer.

We prove that there are infinitely many $(1,1)$-knots which are topologically slice, but not smoothly slice, which was a conjecture proposed by Béla András Rácz.

The paper calculates the slicing degree of knots using advanced homology theories.

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a s…

A new approach simplifies Sliced-Wasserstein distances to improve learning performance.

New lower bound for doubly slice genus using knot signatures.

New proof for some knots being topologically slice.