New examples of positive twisted torus knots with same surface slope found.
problem Finding new examples of positive twisted torus knots with same surface slope.
method Extending Dean's conditions and using new four-parameter family.
result Four-parameter family of positive twisted torus knots with same surface slope.
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-brid…
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yield…
The strong slope conjecture helps identify torus knots.
problem Detecting torus knots using colored Jones polynomials.
method Observation and application of the strong slope conjecture.
result An adequate knot with matching polynomial degrees is a (2,q)-torus knot. Certain torus knots have infinitely many slopes that do not uniquely identify them.
problem Identifying non-characterizing slopes for certain torus knots.
method Applying a condition from Baker and Motegi to show infinitely many non-characterizing slopes.
result The knots T2,2n+3#T−2,2n+1 have infinitely many non-characterizing slopes. For torus knots, non-integer slopes are characterizing.
problem Characterizing slopes for torus knots in 3D space.
method Analysis of p/q-surgery on torus knots using Heegaard Floer homology. result The set of characterizing slopes for torus knots contains all but finitely many non-integer slopes.
Safe screening rule improves Group SLOPE efficiency.
problem Efficiently selecting groups of predictors in high-dimensional sparse learning.
method Safe screening rule for Group SLOPE, addressing block non-separable group effects.
result Significant computational efficiency gains without sacrificing accuracy.
The paper shows how curves converge to a grim reaper with unbounded slopes.
problem Curvature flow with unbounded boundary slopes.
method Uniform interior gradient estimates for symmetric and non-symmetric curves.
result The flow converges to a grim reaper in Cloc2,1((−1,1)imesR) topology. SLOPE optimizes portfolios by grouping similar assets and estimating weights with sorted ℓ1-Norm.
problem Optimizing financial portfolios with sparse selection.
method Sorted ℓ1-Norm penalization (SLOPE) and Alternating Direction Method of Multipliers (ADMM). result SLOPE optimizes portfolios from minimum variance to equally weighted, improving out-of-sample performance.
Solves Fu-Yau equation in arbitrary dimensions and slope.
problem Solving the Fu-Yau equation in arbitrary dimensions and slope.
method Introduced a more stringent ellipticity condition to solve the equation.
result Improved solutions for known and open cases of the Fu-Yau equation.
We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z)≤μ(X) for all subschemes Z. This gives man…
This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the sam…
New knots found that resist trace detection.
problem Knots not detected by any trace.
method Extended existing results to new knots.
result Knots exist that share traces with infinitely many distinct knots and are not detected by any trace.
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect fa…
In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic surgeries. The result is an application of Heegaard Floer theory and number theory.
The Slope Conjecture is verified for a specific family of Montesinos knots.
problem Relating the degree of the colored Jones polynomial to boundary slopes of knots.
method Verification for a specific family of Montesinos knots with given conditions.
result The Slope Conjecture and Strong Slope Conjecture are confirmed for the specified knots.
Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
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.
Study slopes in 3-manifolds, proving conjectures about knots.
problem Understanding slopes in 3-manifolds and their implications for knots.
method Upper bounds on distances between slopes, applications to knots and surgeries.
result Bounds on boundary and degeneracy slopes for knots in 3-manifolds.
Using the Hatcher-Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such knots determine a boundary slope as predicte…
The study identifies characterising slopes for hyperbolic knots and proves stronger results for L-space knots.
problem Characterizing slopes of hyperbolic knots and their properties.
method Combining Lackenby's proof and Heegaard Floer homology genus bounds.
result All but finitely many slopes are characterising for hyperbolic knots and L-space knots.
Constructs Lefschetz fibrations with slopes near 2.
problem Finding Lefschetz fibrations with slopes close to 2.
method Constructs minimal, simply connected genus-g Lefschetz fibrations over the sphere.
result The infimum and supremum of slopes are not achievable.
The construction of knots via annular twisting has been used to create families of knots yielding the same manifold via Dehn surgery. Prior examples have all involved Dehn surgery where the surgery slope is an integral multiple of 2. In this note we prove that for any integer n there exist infinitely many different k…
Study on geometry of mountain slopes as Finsler metrics.
problem Existence and behavior of slope metrics on surfaces of revolution.
method Analysis of Finsler metrics on surfaces of revolution and comparison with Riemannian areas.
result Existence of globally defined slope metrics on surfaces of revolution.
Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Study provides concrete examples of knot slopes.
problem Finding explicit characterizing slopes for knots.
method Concrete examples for the (-2,3,7)-pretzel knot.
result Explicit characterizing slopes for the knot 12n242. 2-level SLOPE improves high-dimensional inference with fewer hyperparameters.
problem High-dimensional linear regression with complex penalty sequences.
method 2-level SLOPE with three hyperparameters, reducing penalty space.
result Sharp trade-off between TPP and FDP, computationally efficient.
The Strong Slope Conjecture is proven for specific knot types.
problem Proving the Strong Slope Conjecture for various knot types.
method Using connect sums and cabling, the conjecture is shown to be closed under these operations.
result The Strong Slope Conjecture is established for graph knots.
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.
problem Model selection in least-squares fixed-design regression with biased models and general noise.
method Proves the slope heuristics works even with significant bias and computes expectations for Gaussian noise.
result The slope heuristics is valid even when models are biased and noise has a general dependence structure.
New slopes identified for torus knots, improving previous results.
problem Identifying characterizing slopes for torus knots.
method Applying recent work on L-space surgeries, improving previous bounds.
result Every nontrivial L-space slope of T5,2 is characterizing for T5,2. New research shows that many slopes are characterizing for satellite knots.
problem Characterizing slopes for satellite knots.
method Detailed examination of the JSJ decomposition of a surgery along a knot, combined with other authors' constraints on surgery slopes.
result Many non-integral slopes are characterizing for composite knots.
A non-trivial slope r on a knot K in S3 is called a characterizing slope if whenever the result of r-surgery on a knot K′ is orientation preservingly homeomorphic to the result of r-surgery on K, then K′ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with $|p| +…
New knots found with specific slope properties.
problem Finding knots with non-integral boundary slopes.
method Kabaya's method for boundary slopes and layered solid torus construction.
result Existence of knots with boundary slopes of any positive denominator.
New rules reduce SLOPE model fitting time by screening out irrelevant variables.
problem Expensive tuning of regularization parameter in penalized regression models.
method Strong screening rules for group-based SLOPE models.
result Significant acceleration of fitting process for Group SLOPE and sparse-group SLOPE.
Paper analyzes SLOPE via AMP, providing an asymptotically sharp analysis and algorithmic approach.
problem Analyzing SLOPE's solution under Gaussian random designs.
method Developed an asymptotically exact characterization using approximate message passing.
result AMP iterates converge to the SLOPE solution in an asymptotic sense.
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-…
Large knots have very varied boundary slopes.
problem Understanding the variability of boundary slopes in knots.
method Analyzing alternating knots and their boundary slopes.
result The ratio of boundary slope diameter to crossing number can be arbitrarily large.
Let X be a norm curve in the SL(2,C)-character variety of a knot exterior M. Let t = || b || / || a || be the ratio of the Culler-Shalen norms of two distinct non-zero classes a, b in H_1(\partial M, Z). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict bound…
Study Mazur doubles of knots and their relation to the Slope Conjecture.
problem Proving the Strong Slope Conjecture for Mazur doubles of knots.
method Analyzing Mazur doubles under certain hypotheses and using the Slope Conjecture.
result Mazur doubles of knots satisfy the Strong Slope Conjecture under certain conditions.
Let r_m and r_M be the least and greatest finite boundary slopes of a hyperbolic knot K in S^3. We show that any cyclic surgery slopes of K must lie in the interval (r_m - 1/2, r_M + 1/2).
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Proves existence of Lefschetz fibrations with arbitrary slopes.
problem Finding Lefschetz fibrations with specific slopes.
method Proves existence for rational numbers in (2,8) with large genus.
result Arbitrary slopes (r in (2,8)) are possible for genus-g Lefschetz fibrations.
This paper completes proofs for left orderable slopes of double twist knots.
problem Proving the left orderability of slopes for double twist knots.
method Continuous families of hyperbolic SL2(R) representations of knot groups.
result Any slope in (-4n, 4m) is a left orderable slope of C(2m+1, -2n).
The paper examines slopes and their norms in exceptional Dehn fillings.
problem Understanding slopes and norms in exceptional Dehn fillings.
method Investigates the relationship between Euclidean length and Culler-Shalen norm on horotori.
result Establishes two inequalities between Euclidean length and Culler-Shalen norm.
New method corrects Laplace/BIC errors in singular models, revealing effective dimension.
problem Laplace/BIC errors in singular models due to incorrect effective dimension assumption.
method RLCT (real log canonical threshold) to correct effective dimension in linear models.
result Correct evidence slope and effective dimension estimation in linear settings.