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Knot exterior

Definition

Assume countable choice, as required for the smooth normal-bundle structure in the tubular-neighbourhood interface. Let $K : S^{1} \to S^{3}$ be a smooth oriented knot and let $ν (K) = T S^{3} ∣_{K} / T K$ be its quotient normal bundle. Equip $ν (K)$ with a chosen smooth fibre norm $∥ \cdot ∥$ . Let $Φ : Ω \to S^{3}$ be a tubular-neighbourhood map from an open neighbourhood $Ω$ of the zero section in $ν (K)$ that restricts to the inclusion of $K$ . For $ϵ > 0$ , define the closed disk subbundle

D_{ϵ} ν (K) : = {v \in ν (K) : ∥ v ∥ \leq ϵ} .

Choose $ϵ$ so that $D_{ϵ} ν (K) \subseteq Ω$ , and set $N (K) = Φ (D_{ϵ} ν (K))$ . The fibre norm, tubular map, and radius are part of the chosen tubular data. The exterior of this chosen tube is

E_{K} : = S^{3} ∖ {int}_{S^{3}} N (K) .

Retain its boundary $T_{K} : = \partial E_{K} = \partial N (K)$ , orient $E_{K}$ by restriction of the fixed orientation of $S^{3}$ , and orient $T_{K}$ as the boundary of $E_{K}$ . The choice of tubular map and radius is part of the input to this definition.

Definition references

Cattabriga and Francaviglia, §4.1

Hatcher, Theorem 3.44 and Corollary 3.45

Depends on

Used by

A zero Wada denominator makes that deleted column inadmissible Counterexample
Abelianized peripheral data do not distinguish the unknot from the trefoil Counterexample
An arbitrary parallel push-off need not be preferred Counterexample
Equal surgery manifolds need not have isotopic diagrams Counterexample
The trefoil exterior is Seifert-fibered Counterexample
Trefoil surgery with trivial first homology and nontrivial fundamental group Counterexample
Adjoint-twisted homology and Culler-Shalen data Definition
Based string links and labeled link peripheral systems Definition
Boundary slope of an incompressible surface in a knot exterior Definition
Candidate instanton surgery-triangle input data Definition
Coprime torus knot Definition
Cyclic branched cover of a knot Definition
Finite-volume hyperbolic knot exterior Definition
Incompressibility, Seifert fibrations, and JSJ systems Definition
Instanton knot homology and its canonical isomorphism class Definition
Integral framed-link surgery diagram Definition
Meridian and preferred longitude Definition
Oriented slopes and knot Dehn filling Definition
Satellite pattern, companion, and winding number Definition
Singular instanton knot and sutured closure data Definition
SL2 character variety and peripheral eigenvalues Definition
SL2(C)-abelian surgery Definition
Smooth zero-trace of a knot Definition
The admissible Wada quotient of a deficiency-one exterior spine Definition
The maximal-abelian-cover Alexander module of an oriented link Definition
The ordinary knot A-polynomial Definition
Topological linking number as a meridian coordinate Definition
Twisted chain complex and based Reidemeister torsion for link exteriors Definition
Zero-framed surgery on a knot Definition
Trefoil torsion from its torus spine and colored Burau block Example
Worked case: the figure-eight complement Example
Worked case: the unknot exterior Example
Worked cases of three-manifold surgery presentations Example
A knot exterior has first homology generated by its meridian Lemma
A nonseparating incompressible longitudinal surface in a fibered exterior is a fiber Lemma
A smooth knot exterior is irreducible Lemma
An S3 filling detects the meridian slope Lemma
Analytic foundations for the singular instanton knot complex Lemma
Atoroidal non-Seifert knot exteriors have the required topological hypotheses Lemma
Baldwin–Sivek instanton-surgery and Newton-polygon input module Lemma

…and 30 more results.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes (standard reference, not scraped)
Allen Hatcher, Algebraic Topology (standard reference, not scraped)