A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…
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48 results for “Seifert”
How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the ot…
We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.
We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a na…
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…
We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…
This paper constructs Seifert-fibered Dehn surgeries for hyperbolic tunnel-number-one knots.
problem Constructing Seifert-fibered Dehn surgeries for hyperbolic tunnel-number-one knots.
method Using disjoint curves in a genus two handlebody, the paper constructs Seifert-fibered Dehn surgeries.
result Seifert-fibered Dehn surgeries on hyperbolic tunnel-number-one knots can arise from primitive/Seifert positions.
We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resu…
Extends Seifert algorithm to 3-manifolds via surgery.
problem Construct Seifert surfaces in arbitrary 3-manifolds.
method Uses surgery on framed links in S^3 to extend classical algorithm.
result Explicit construction of Seifert surfaces in 3-manifolds.
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…
Seifert-Tait graphsmath.GT
Dominant knots have isomorphic Seifert and Tait graphs.
problem Understanding knot dominance through graph isomorphism.
method Examined alternating knots and their Seifert and Tait graphs.
result Isomorphic Seifert and Tait graphs indicate dominant knots.
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.
problem Analyzing the non-uniqueness of Seifert fibrations in spherical 3-orbifolds.
method Examined closed spherical Seifert three-orbifolds, determining the number and describing algorithms for equivalence.
result Determined the number of inequivalent fibrations for any closed spherical Seifert three-orbifold.
Classifies generalized Seifert fiber spaces and their branched covers.
problem Classifying generalized Seifert fiber spaces and their topological properties.
method Symbolic classification and computation of invariants.
result Canonical double branched cover of non-manifold spaces is a Seifert manifold.
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
We define the surface complex for -manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex is…
We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and ar…
Paper constructs infinitely many pairs of Seifert surfaces for each link.
problem Constructing infinitely many pairs of Seifert surfaces for each link.
method Using a multiple group rack (MGR) to construct invariants and distinguishing surfaces using these invariants.
result Presented infinitely many pairs of Seifert surfaces for each link, satisfying specific conditions.
Researchers calculate the volume of Seifert representations for graph manifolds and their covers.
problem Computing the volume of Seifert representations for graph manifolds and their finite covers.
method Established an effective formula for computing the volume of Seifert representations of graph manifolds and obtained restrictions analogous to the Milnor–Wood inequality.
result The Seifert volume of any graph manifold is a rational multiple of π², and the supremum ratio of the Seifert volume over the covering degree can be positive or infinite.
We introduce and define "oriented framed measured lamination links" in a 3-manifold . These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links boun…
Method to create rational Seifert surfaces for knots in Lens space.
problem Creating rational Seifert surfaces for knots in Lens space.
method Assuming a regular projection, construct rational Seifert surface on twist toroidal diagram.
result A method to construct rational Seifert surfaces for knots in Lens space.
Corrects classification of Seifert fibrations for lens spaces with non-orientable bases.
problem Classifying Seifert fibrations of lens spaces with non-orientable bases.
method Corrected an earlier lemma and filled a classification gap.
result Corrected the classification of Seifert fibrations for lens spaces with non-orientable bases.
We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.
problem Understanding the asymptotic behavior of Turaev-Viro invariants for Seifert fibered 3-manifolds.
method Analysis of large asymptotic behavior of Turaev-Viro invariants.
result Proved the volume conjecture for Seifert fibered 3-manifolds with empty and non-empty boundaries.
Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix…
The paper defines a function for knots in Seifert manifolds and connects it to Witten-Reshetikhin-Turaev invariants.
problem Defining a function for knots in Seifert manifolds.
method Explicit construction of a function Φ(q; N) and its properties.
result The function Φ(q; N) satisfies a q-difference equation related to character varieties.
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
Study flippable Heegaard splittings in Seifert fibered spaces.
problem Identifying flippable Heegaard splittings in Seifert fibered spaces.
method Examining isotopies that interchange Heegaard splitting sides in Seifert fibered spaces.
result Characterized which Heegaard splittings are flippable in Seifert fibered spaces.
Algorithm calculates Seifert matrices for colored links.
problem Computing Seifert matrices for colored links.
method Developed an algorithm implemented in Clasper software.
result Computes Seifert matrices, potential function, and signatures.
In a paper published in 2002, the author gave a criterion to determine whether there is a fiber-preserving branched covering between two given orientable Seifert manifolds with orientable bases. Here we supply some details of the proof of two claims in that paper. We give an explicit construction of fiber-preserving br…
We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S^3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S^3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic p…
New method alters Seifert surfaces without compression.
problem Understanding alterations of Seifert surfaces.
method Introducing alteration as a new surface operation.
result Seifert surfaces can be altered without compression.
Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.
problem Characterize essential surfaces in Seifert fiber spaces with singular surfaces.
method Extends Frohman and Rannard's approach to handle surfaces with singular fibers.
result Characterizes essential surfaces in Seifert fiber spaces with singular surfaces.
2-knot manifolds have Seifert fibered base orbifolds.
problem Characterizing Seifert fibered 2-knot manifolds.
method Analyzing aspherical 2-orbifolds with orbifold fundamental groups of weight 1.
result Aspherical 2-orbifolds with orbifold fundamental groups of weight 1 are Seifert fibered bases of 2-knot manifolds.
Researchers found a way to measure the complexity of Seifert fibered spaces with boundaries.
problem Measuring the complexity of Seifert fibered spaces with boundaries.
method Relating triangulation complexity to Seifert data and using barycentric subdivision.
result Determined triangulation complexity in terms of Seifert data and showed singular fibres can be made simplicial.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
Recently Kearton showed that any Seifert matrix of a knot is S--equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S--equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by appl…
New method constructs Seifert solids from bridge trisections.
problem Constructing Seifert solids from bridge trisections.
method Adapting Seifert's algorithm to tri-plane diagrams.
result Classification results on surface decomposability and unknottedness.
3D topological order linked to Seifert manifolds and gauge groups.
problem Classifying 3D topological orders using Seifert manifolds and gauge groups.
method Correspondence between topological order, Seifert manifolds, and ADE gauge groups.
result Construction of modular fusion categories from Seifert manifolds and gauge groups.
Recognition of Seifert fibered spaces with boundary is computationally tractable.
problem Recognizing Seifert fibered spaces with boundary.
method Proving the existence of fundamental horizontal surfaces and normal vertical annuli with bounded total weight.
result The decision problem is in NP and the construction problem is in FNP.
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of exa…
New non-isotopic Seifert surfaces found in 4-ball.
problem Finding non-isotopic Seifert surfaces for knots and links.
method Using double branched covers and Seifert forms, with an algorithm based on quadratic forms.
result Families of knots and links with non-isotopic Seifert surfaces in .
Classifies tight contact structures on specific Seifert fibered manifolds.
problem Classifying tight contact structures on Seifert fibered manifolds.
method Constructed contact structures using Legendrian surgery and used convex surface theory for the upper bound.
result Found the lower and upper bounds for tight contact structures.
If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.
Study on Seifert 4-manifolds to classify profinite rigidity.
problem Understanding profinite completions of Seifert 4-manifolds' fundamental groups.
method Classification of monodromies and conditions for profinite rigidity.
result Classification of Seifert 4-manifolds based on their profinite completions.
We study some properties of transverse contact structures on small Seifert manifolds, and we apply them to the classification of tight contact structures on a family of small Seifert manifolds.
Virtual Seifert Surfacesmath.GT
A virtual knot that has a homologically trivial representative in a thickened surface is said to be an almost classical (AC) knot. then bounds a Seifert surface . Seifert surfaces of AC knots are useful for computing concordance invariants and slice ob…