In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to…
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48 results for “Casson-Gordon Rectangle Condition”
The study connects Dehn surgeries on links to trisections of 4-manifolds.
problem Understanding Dehn surgeries on links via trisections.
method Expanding known connections and introducing new trisection conditions.
result New trisection conditions can obstruct reducibility of trisections.
Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.
problem Detecting strong irreducibility from the rectangle condition.
method Constructing a double branched cover of a knot in S^3.
result Found a genus 2 Heegaard splitting that is strongly irreducible but fails rectangle condition.
In this paper, we define the rectangle condition on the bridge sphere for a -bridge decomposition of a knot whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of -manifolds. We show that the satisfaction of the rectangle condition for a -bridge decomposition can …
We give the rectangle condition for strong irreducibility of Heegaard splittings of -manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a -fold covering of branched along a link. The condition implies that any thin meridional level surface in the link complement is incom…
The paper proves geometric properties of square tables and saddle surfaces.
problem The mathematical table problem from a geometric-topological perspective.
method Geometric-topological proofs on cylinder, saddle surfaces, and level sets of Fenn graphs.
result Zero-existence theorem on a cylinder, proving Fenn's square-table theorem under different boundary conditions.
Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
We show that Scharlemann-Thompson untelescoping of Heegaard splittings is finer than Casson-Gordon's by giving concrete examples.
It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show …
Lower bound on stable 4-genus of knots using Casson-Gordon signatures.
problem Finding a lower bound on the stable 4-genus of knots.
method Using Casson-Gordon τ-signatures to compute the lower bound.
result A twist knot is torsion in the knot concordance group if and only if it has vanishing stable 4-genus.
The paper studies twisted signature invariants and their relation to Casson-Gordon invariants.
problem Obstructions to knot concordance using twisted signature invariants.
method Defining a twisted signature function σ_{K,ρ} and proving satellite formulas.
result The twisted signature function σ_{K,ρ} is closely related to Casson-Gordon invariants for appropriate metabelian representations.
Optimal weight windows are symmetric rectangles centered at peak.
problem Finding the best weight windows for weighted least squares.
method Investigated symmetric and tapered rectangle window weights, showing the best rectangle window is optimal.
result The best rectangle window is optimal for all tapered rectangle window definitions.
Curves inscribe rectangles with positive area.
problem Finding angles for inscribing rectangles within Jordan curves.
method Proving existence of a subset of angles with measure at least A/R^2.
result Angles inscribing rectangles have measure at least A/R^2.
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group.
The study describes a cell structure for multisets in a rectangle.
problem Understanding the space of multisets in a rectangle.
method Developed a piecewise Euclidean bi-simplicial cell structure.
result Connected to spaces of complex polynomials and permutahedra.
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge kno…
The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.
problem Homological stability in the space direction of square configurations.
method Analyzing the ordered configuration space of squares in a rectangle.
result Most rectangles can be almost entirely filled with squares and still have stable homology.
The Rectangular Peg Problemmath.GT
Rectangles can fit on smooth curves, proving a theorem about Klein bottles.
problem Fitting rectangles on smooth curves.
method Shevchishin's theorem about Klein bottle embeddings.
result Similar rectangles can be placed on smooth Jordan curves.
Paper classifies pillow box isometric deformations preserving crease patterns.
problem Classifying pillow box isometric deformations preserving crease patterns.
method Continuous isometric deformations from pillow boxes to double rectangles, preserving crease patterns.
result Such deformations necessarily change pillow box topology.
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…
Characterizes discrete nets with characteristic properties on larger parameter rectangles.
problem Classify discrete and smooth surfaces with specific properties.
method Imposes characteristic properties on larger parameter rectangles for nets.
result Characterizes discrete multi-nets leading to classical smooth surfaces.
We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.
In this paper it is shown that manifolds admitting minimal genus weakly reducible but irreducible Heegaard splittings contain an essential surface. This is an extension of a well known theorem of Casson-Gordon to manifolds with non-empty boundary. The situation for non-minimal genus Heegaard splittings is also investig…
A new ensemble model uses simple hyper-rectangles to improve gradient boosting machine performance.
problem Improving gradient boosting machine performance and avoiding overfitting.
method Proposes a new ensemble model with axis-parallel hyper-rectangles as base models, integrates into GBM, and uses SHAP for interpretation.
result GBM with HRBMs can be an effective and interpretable model for regression and classification problems.
Any cyclic quadrilateral can fit inside any smooth curve.
problem Inscribing cyclic quadrilaterals in smooth curves.
method Proving geometric properties of cyclic quadrilaterals and smooth curves.
result Cyclic quadrilaterals can be inscribed in any closed convex -curve.
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.
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.
Equal diagonal energies proven on Liouville surfaces.
problem Diagonal energies on Liouville surfaces.
method Analyzing parameter curves and rectangles on Liouville surfaces.
result Diagonal energies are equal in n-dimensional Liouville manifolds.
Invariants of -knotsmath.GT
Survey of invariants for knotted 2-spheres in 4-space.
problem Characterizing knotted 2-spheres in 4-dimensional space.
method Algebraic topology of knot exterior, gauge theory, combinatorial methods.
result Details are scarce and new results inexistent.
We give a sufficient condition under which vanishing property of Cochran-Orr-Teichner knot concordance obstructions splits under connected sum. The condition is described in terms of self-annihilating submodules with respect to higher-order Blanchfield linking forms. This extends results of Levine and the authors on di…
In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, w…
The study shows that inscribed rectangles in smooth curves cover at least one third of all possible aspect ratios.
problem Determining the coverage of inscribed rectangles in smooth Jordan curves.
method Analyzing sets of disjoint homologically nontrivial projective planes and applying Kemperman's theorem.
result The Lebesgue measure of the set of aspect ratios is at least 1/3.
We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.
For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched covers of the knot, obstruct the smooth sliceness of the knot. These invariants bear …
ARGEN method improves variable selection and regularization in high-dimensional sparse models.
problem Constrained variable selection and regularization in high-dimensional sparse linear models.
method ARGEN penalty method, variable selection and regularization.
result ARGEN method has variable selection and estimation consistency under certain conditions.
Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of is isomorphic to that of K' as Z[Z] modules. Let K be a 2-knot which is ribbon-move equivalent to the trivial knot. T…
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.
problem Understanding free path lengths on flat surfaces with circular obstacles.
method Proved the existence of a limiting distribution using radius of obstacles as a parameter.
result Relates the limiting distribution to heights of zippered rectangle decompositions.
The paper computes a pairing for knot concordance and finds non-slice knots.
problem Computing and understanding the twisted Blanchfield pairing for knot concordance.
method Combinatorial algorithm for computing the twisted Blanchfield pairing; using zero-framed surgery and Casson-Gordon representations.
result Some satellites of genus two ribbon knots are non-slice.
New integral defined for Hölder continuous functions, characterizing distributional volume forms.
problem Defining and characterizing a new integral for Hölder continuous functions.
method Constructing a distribution from Hölder continuous functions and using integral properties.
result Characterizes the Hölder regularity of the constructed distribution.
The width-volume inequalitymath.DG
We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…
We show that bilinear cup products with local coefficients of closed 3-manifolds recover some twisted pairings of infinite covers and the Casson-Gordon local signatures. As a result, we further give diagrammatic computations of the pairings and signatures.
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was ca…
Fast BATLLNN speeds up verification of TLL NNs by 400x.
problem Verifying output constraints for TLL NNs.
method Uses TLL architecture and decoupled box constraints to improve verification performance.
result 400x faster than state-of-the-art verifiers.