arXiv research

We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $α$-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-…

Study shows existence of Strominger system solutions is not stable under complex structure deformations.

New examples of positive twisted torus knots with same surface slope found.

Paper defines new stability and metrics for complex spaces.

New knots found with specific slope properties.

We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. Afte…

Proves rational slopes characterize knot 5_2, except for integers.

We show that, for any positive real number, there exists a knot in the 3-sphere admitting a pair of boundary slopes whose difference is at most the given number.

New surgery exact triangles in Heegaard Floer homology for rational slopes.

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…

2-level SLOPE improves high-dimensional inference with fewer hyperparameters.

This paper constructs Seifert-fibered Dehn surgeries for hyperbolic tunnel-number-one knots.

The paper finds left orderable slopes for specific double twist knots.

In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pi…

We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" Donaldson-Futaki invariants implies K-stability (resp. K-semistability). We show its applicability…

The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. We generalize their construction and calculate th…

The paper characterizes SLOPE's trade-off between FDP and TPP, showing its power limit and superiority over Lasso.

Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.

The paper finds a surprising positive correlation between upstreamness and downstreamness in global value chains.

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we generalized their constru…

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0…

New proof for a knot type not admitting certain surgeries.

The Euler class of taut foliations on Dehn fillings is studied, leading to new left-orderable slopes.

Paper surveys balanced metrics and proves a geodesic convexity result.

New non-holomorphic Lefschetz fibrations with $(-1)$-sections found.

The level crossing and inverse statistics analysis of DAX and oil price time series are given. We determine the average frequency of positive-slope crossings, $ν_α^+$, where $T_α =1/ν_α^+$ is the average waiting time for observing the level $α$ again. We estimate the probability $P(K, α)$, which provides us the probab…

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive …

The Slope Conjecture is verified for a specific family of Montesinos knots.

The simplest hyperbolic knot has no integral characterizing slopes.

Proves conjectures for pretzel knots using polynomial degrees.

The study confirms a conjecture for a specific type of knot.

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.

Jones slopes characterize adequate knots under the Strong Slope conjecture.

The Links-Gould polynomial of alternating knots is shown to be log-concave and positive.

Study slopes in 3-manifolds, proving conjectures about knots.

Bayesian SLOPE combines SLOPE with Bayesian methods for linear regression.

The study identifies characterising slopes for hyperbolic knots and proves stronger results for L-space knots.

Constructs Lefschetz fibrations with slopes near 2.

Study on geometry of mountain slopes as Finsler metrics.

Paper analyzes origami slope gaps and their distribution, finding a unique pattern.

Study provides concrete examples of knot slopes.

The Strong Slope Conjecture is proven for specific knot types.

Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the m…

Rejoinder on slope heuristics for model selection in regression.

New slopes identified for torus knots, improving previous results.

New research shows that many slopes are characterizing for satellite knots.