Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-pl…
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48 results for “Hopf plumbing”
Study Hopf bands on specific Seifert surfaces, linking to Coxeter groups.
problem Classify Hopf bands on arborescent Seifert surfaces.
method Analyze Hopf bands up to homology and monodromy action.
result Finite set of Hopf bands related to Coxeter groups.
Let be an oriented link with an alternating diagram . It is known that is a fibered link if and only if the surface obtained by applying Seifert's algorithm to is a Hopf plumbing. Here, we call a Hopf plumbing if is obtained by successively plumbing finite number of Hopf bands to a disk. In t…
Trefoil Plumbingmath.GT
We give a criterion for an open book to contain an n-times iterated Hopf plumbing summand. As an application, we show that fibre surfaces of positive braid knots admit a trefoil plumbing structure.
Fibre surfaces in 3-sphere can have unbounded stabilisation height.
problem Understanding the stabilisation height of fibre surfaces in 3-sphere.
method Using Hopf plumbing operations and Stallings twists.
result Families of fibre surfaces can have unbounded stabilisation height.
By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divid…
We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the …
Homogeneous braids are visually prime, solving a Cromwell question.
problem Addressing Cromwell's question about visually prime braids.
method Using a criterion for primeness of open books and Murasugi sums.
result Closures of homogeneous braids are visually prime.
Authors show that genus defects of Hopf arborescent links are decidable.
problem Computing the genus defects of four-dimensional knots.
method Proving a well-quasi-order of Seifert surfaces under a relation of minors.
result Decidability of computing genus defects for Hopf arborescent links.
Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteris…
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in .
method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., , , , and …
Open books link prime positive braids to checkerboard graphs.
problem Determining link types of prime positive braids.
method Associated open books with checkerboard graphs.
result Link type of a prime positive braid closure is determined by the linking graph.
New findings show almost all strongly quasipositive braids can be fibered.
problem Understanding which strongly quasipositive braids can be fibered.
method Using the Birman-Ko-Lee presentation and geometric plumbing techniques.
result All closures of strongly quasipositive braids with certain conditions are fibered.
New links identified with Alexander polynomial signs to detect satellite links.
problem Detecting satellite links using Alexander polynomial coefficients.
method Introduced a set of links including positive braids and arborescent Hopf plumbings. Showed links in this set have opposite signs for leading and second coefficients of Alexander polynomials.
result Certain satellite links, like (n,1)-cables, are not in the set of links with opposite sign coefficients.
Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight …
Basket links are shown to be isotopic to .
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to matrix.
result Basket links are isotopic to .
We define the basket number, the flat plumbing number and the flat plumbing basket number of a link. Then we provide some upperbounds for these plumbing numbers by using Seifert's algorithm. We study the relation between these plumbing numbers and the genera of links.
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain…
Study fibered alternating links using Coxeter transformations.
problem Understanding the Alexander polynomial and monodromy of fibered links.
method Alternating-sign Hopf plumbing and Coxeter transformations.
result Strong support for Hoste's conjecture and bi-orderability of link groups.
Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the…
Study connects plumbing baskets to 3-sphere contact structures.
problem Understanding contact structures in 3-sphere.
method Examine flat plumbing baskets and their relation to contact structures.
result Lower bounds and specific numbers for various knots and links.
Two special moves relate any two open book decompositions of a 3-manifold.
problem Relating any two open book decompositions of a 3-manifold.
method Using 4-dimensional Lefschetz fibrations and 3-dimensional contact topology, including Harer's twisting and Giroux-Goodman stable equivalence theorem.
result A complete set of two moves suffices to relate any two open book decompositions of a given 3-manifold.
The existence of basket, flat plumbing and flat plumbing basket surfaces of a link was first proven from a braid representative of the link. In the present article, we show the existence of such surfaces from an induced graph of the link. Consequently, we define the basket number, flat plumbing number and flat plumbing…
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. In present article, we study links of the flat plumbing basket numbers or less using a special presentation of the flat plumbing basket surfaces. We find a complete classification theorem of links of the fla…
The paper constructs a Legendrian link from a flat plumbing basket and relates its properties to the self-linking and Thurston-Bennequin numbers.
problem Understanding the properties of Legendrian links constructed from flat plumbing baskets.
method Constructing a Legendrian link from a flat plumbing basket and analyzing its self-linking and Thurston-Bennequin numbers.
result Determining the flat plumbing basket numbers of torus links.
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
New invariant for plumbed 3-manifolds with detailed computations and conjectures.
problem Computing and understanding invariants for plumbed 3-manifolds.
method Introducing a two-variable refinement , analytically computing limits, and proposing conjectures based on numerical data.
result Proposed conjecture that the recovered is an invariant for all tree plumbed 3-manifolds.
New quantum modular forms for specific 3-manifolds discovered.
problem Computing WRT invariants of plumbed 3-manifolds.
method Investigating -series at radial limits for unimodular plumbing graphs.
result Proving is a depth two quantum modular form.
Algorithm transforms weakly negative plumbing trees to negative definite ones.
problem Transforming weakly negative plumbing trees to negative definite ones.
method Combining plumbing calculus with diagonalization algorithm to systematically eliminate positive eigenvalues.
result Explicit algorithm reduces weakly negative definite plumbing trees to negative definite ones.
Proves a plumbing-multiplicative property of a Links-Gould invariant.
problem Proving a multiplicative property of the Links-Gould invariant.
method Using Laurent polynomials and topological plumbing of surfaces.
result The Links-Gould invariant is monic in the Laurent polynomial ring.
Study homology cobordism and plumbed manifolds using Heegaard Floer homology.
problem Understanding homology cobordism and local equivalence between plumbed manifolds.
method Use involutive Heegaard Floer homology to analyze linear combinations of almost-rational plumbed three-manifolds.
result Proves that certain plumbed manifolds are not torsion in the homology cobordism group.
We consider double plumbings of two disk bundles over spheres. We calculate the Heegaard-Floer homology with its absolute grading of the boundary of such a plumbing. Given a closed smooth 4-manifold and a suitable pair of classes in , we investigate when this pair of classes may be represented by a config…
Plumbing of surfaces embeds in hyperbolic 4-manifolds.
problem Embedding specific geometric structures in hyperbolic manifolds.
method Using plumbings of disc bundles over surfaces.
result Plumbed structures can be embedded in closed hyperbolic 4-manifolds.
New symplectic operations replace certain 4-manifolds with simpler ones.
problem Symplectic replacement of complex 4-manifolds.
method Introducing new symplectic cut-and-paste operations to replace plumbings with simpler Euler characteristic 2 4-manifolds.
result Constructed a symplectic exotic using a 2-replaceable plumbing tree.
New invariant connects knot homology and BPS series for plumbed knot complements.
problem Understanding invariants of plumbed knot complements.
method Introducing an invariant unifying knot lattice homology and BPS series, proving a surgery formula.
result Proved a surgery formula relating the new invariant to the weighted graded root of the surgered 3-manifold.
Study series invariants of plumbed 3-manifolds using root lattices.
problem Understanding invariants of plumbed 3-manifolds twisted by root lattices.
method Use formal series to study invariants, decompose , and compute in specific cases.
result Show that is unique and decomposes into related series invariant under five Neumann moves.
New method for calculating crosscap numbers of knots.
problem Calculating crosscap numbers of knots efficiently.
method Deformation of plumbing and computer aid.
result Efficient calculation of crosscap numbers.
New invariants measure how far spanning surfaces are from being compressible.
problem Understanding how essential spanning surfaces are in 3-manifolds.
method Introducing algebraic and geometric essence invariants, proving plumbing respects algebraic essence, and extending results to arbitrary 3-manifolds.
result Plumbing respects the algebraic essence of spanning surfaces, extending Ozawa's theorem.
Study tight contact structures on specific 3-manifolds.
problem Counting and constructing tight contact structures on plumbed 3-manifolds.
method Algorithm to construct stein diagrams for tight structures without Giroux torsion.
result Explicit algorithm to construct tight contact structures.
This paper classifies minimal fillings of lens spaces.
problem Determining the smallest smooth negative-definite fillings of lens spaces.
method Classification of minimal fillings based on forbidden subgraphs in plumbing graphs.
result Classification of lens spaces with minimal negative-definite canonical plumbing.
Study on invariants of plumbed 3-manifolds, comparing with Heegaard-Floer homology.
problem Analyzing invariants of plumbed 3-manifolds.
method Expressing of Seifert manifolds in terms of singularity theory invariants, providing examples for non-Seifert manifolds, comparing with Heegaard-Floer homology.
result Interesting behavior of invariants for non-Seifert manifolds, comparison with Heegaard-Floer homology.
Study properties of contact structures on symplectic disk bundles with concave boundaries.
problem Understanding the geometric properties of contact structures on concave boundaries of symplectic disk bundles.
method Use tools from toric geometry and algebraic torsion measurements from embedded contact homology.
result All such contact manifolds have a global contact toric structure, and can be tight or overtwisted.
Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytopes from the plumbing graphs toget…
Algorithm calculates polynomial Seiberg-Witten invariant from zeta-function of plumbed 3-manifolds.
problem Computing the Seiberg-Witten invariant for plumbed 3-manifolds.
method Némethi's division algorithm for zeta-functions.
result Explicit calculation of polynomial Seiberg-Witten invariant.
Study series invariants for plumbed 3-manifolds and their properties.
problem Understanding series invariants for plumbed 3-manifolds and their applications.
method Twisted root lattice, gluing and splitting properties, explicit description of lens spaces and Brieskorn spheres.
result Series verify gluing and splitting properties of 3-manifolds.
On T-positive linksmath.GT
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
Study Heegaard Floer homology of plumbed 3-manifolds, proving infinite-rank subgroup existence.
problem Computing Heegaard Floer homology for specific 3-manifolds.
method Involutive Heegaard Floer homology calculations.
result Existence of an infinite-rank subgroup in the homology cobordism group.