New techniques simplify computing geometric index of links in solid tori.
problem Computing the geometric index of links in solid tori.
method Introducing geometric index for solid chambers and proving its computation under specific conditions.
result Simplified computation of geometric index for new examples.
The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.
problem Existence of meridional essential surfaces in knot exteriors.
method Analyzing knot exteriors and their surfaces.
result Existence of infinitely many knot exteriors with meridional essential surfaces of any genus and boundary components.
We determine all (1,1)-knots which admit an essential meridional surface, namely, we give a construction which produces (1,1)-knots having essential meridional surfaces, and show that if a (1,1)-knot admits an essential meridional surface then it comes from the given construction.
The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
Whitehead doubles have matching meridional rank and bridge number.
problem Determining the meridional rank of Whitehead doubles.
method Analyzing algebraically tame knots and their Whitehead doubles.
result Meridional rank and bridge number coincide for Whitehead doubles of prime knots.
Study proves meridional rank conjecture for certain complex links.
problem Determining the minimum number of generators needed for link groups.
method Using Coxeter quotients and Wirtinger numbers, the study provides both lower and upper bounds.
result Proved meridional rank conjecture for specific types of links.
The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.
problem Investigating the Meridional Rank Conjecture for knotted surfaces and welded knots.
method Constructing knots with specific properties, establishing equalities, and using Tube map.
result Established the equality of bridge number and meridional rank for certain knots and knotted spheres.
We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and meridional ranks of links. To prove this, we also show that the meridional rank of any sa…
Algorithm computes meridional knot summands in Dehn surgeries.
problem Computing meridional summands in Dehn surgeries for knots.
method Introduced bordered Floer bimodule algorithm for knot complements and meridians.
result Grading of the module computes meridional knot summands in Dehn surgeries.
New prime knots found with high-genus surfaces.
problem Finding knots with high-genus surfaces.
method Analyzing prime knot complements for meridional essential surfaces.
result Existence of infinitely many prime knots with high-genus surfaces.
Adding an unknot to any link equals its bridge number and meridional rank.
problem Proving the Meridional Rank Conjecture for any link.
method Embedding an unknot in a link's complement to achieve the conjecture and proving it for new families of links.
result Bridge numbers and meridional ranks are equal for any link and its unknot.
Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.
problem Finding maximal rank Coxeter quotients for knots.
method Computational approach to find Coxeter quotients for knots up to 16 crossings.
result Verification of Meridional Rank Conjecture for 595,515 knots.
We prove for a large class of knots that the meridional rank coincides with the bridge number. This class contains all knots whose exterior is a graph manifold. This gives a partial answer to a question of S. Cappell and J. Shaneson, see problem 1.11 on Kirby's list.
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …
We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disj…
We show that, for any integer n≥3, there is a prime knot k such that (1) k is not meridionally primitive, and (2) for every m-bridge knot k′ with m≤n, the tunnel numbers satisfy t(k#k′)≤t(k). This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel number under connected sum…
We define Wirtinger number for links and prove it equals bridge number.
problem Determining the minimum number of generators for knot groups.
method Defining Wirtinger number and proving it equals bridge number.
result Wirtinger number equals bridge number for links.
We give a description of all (1,2)-knots in S^3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)-knots having a meridionally incompressible surface of genus 2, and then show that any such surface for a (1,2)-knot must come from one …
The study proves a conjecture about arborescent links with many twigs.
problem Proving the meridional rank conjecture for arborescent links.
method Using an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree.
result Proves the meridional rank conjecture for arborescent links with specific properties.
We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M th…
New examples show high tree-width in knot diagrams.
problem Understanding the tree-width of knot diagrams.
method Induced sphere-decompositions and essential surfaces.
result First examples of high tree-width knots.
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
The study refines knot groups and creates a metric Gordian graph filtration.
problem Understanding the structure and relationships of knots.
method Defining a metric filtration of the Gordian graph based on surjective symmetric group quotients.
result Verification of the Meridional Rank Conjecture for knots with specific properties.
The study bounds exceptional surgeries for hyperbolic knots.
problem Identifying the range of slopes for exceptional surgeries.
method Analyzing meridional and non-meridional surgeries, and investigating the relationship between boundary slopes and exceptional surgeries.
result There are boundary slopes b1<b2 such that all non-trivial exceptional surgeries occur in the interval [b1,b2]. The integers in $[\ceil{b_1}, \floor{b_2}]$ are all exceptional surgeries. If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the A-polynomial of the knot. For example, we show that for 2-bridge knots the polynomial…
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
Link's sphere number equals its bridge number.
problem Determining the minimum number of generators for a link's fundamental group.
method Using meridional presentations with embedded two-spheres in fixed diagrams.
result The minimum number of generators equals the bridge number.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies Γ-homotopy ribbon slice discs up to topological ambient isotopy. result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs. New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
New findings on L-spaces and taut foliations in hyperbolic links.
problem Characterizing L-spaces and taut foliations in Dehn surgeries on hyperbolic links. method Analyzing rational homology spheres and using coorientable taut foliations.
result Non-meridional surgeries on fibered hyperbolic two-bridge links support coorientable taut foliations.
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.
We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and …
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
Study geodesic discs with boundary length bounds, finding their closure in metric space.
problem Geodesic discs with boundary length constraints in metric spaces.
method Investigate closure in Gromov-Hausdorff space, relate to disc retracts.
result Closure of geodesic discs is related to disc retracts in metric spaces.
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
Study smooth manifolds using disc-presheaves.
problem Understanding smooth manifolds.
method Using presheaves on a category of discs.
result Disc-presheaves have desirable properties and strong applications.
This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n-gon. result Characterization of monohedral tilings of any regular n-gon with up to three tiles. Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.
The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase Δg found using Baumkuchen lemma.