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Every smooth oriented knot in bounds a Seifert surface
Statement
Assume the Axiom of Countable Choice. Every smooth oriented knot bounds a connected compact smooth oriented embedded surface whose oriented boundary is .
Statement references
Facts & Assumptions
Given: and a smooth oriented knot .
A link in the standard may be moved by an orientation-preserving ambient isotopy to lie in . (Oriented links in the three-sphere and ambient isotopy)
Under , every oriented link contained in has a dense set of regular projection directions. (Existence of regular projections)
A regular oriented diagram of one knot component has a connected compact smooth oriented embedded disk-and-band realization with that oriented boundary. (Seifert disks and bands form an embedded oriented spanning surface)
The Axiom of Countable Choice is the sequence principle stated in the item. (The Axiom of Countable Choice ())
Proof
By [F1], move by an orientation-preserving ambient isotopy into . Apply [F2] and [A1] to choose a regular projection direction. The resulting decorated projection is a finite regular oriented diagram of a single knot.
Apply [F3] to to obtain a connected compact smooth oriented embedded surface with oriented boundary . Pull the surface back by the inverse ambient isotopy from step 1.1; the inverse diffeomorphism preserves embeddedness, compactness, connectedness, orientation, and the boundary orientation, giving the required for the original .
Depends on
Used by
- The knot Alexander polynomial multiplies under connected sum Corollary
- Seifert surfaces and smooth knot genus Definition
- Smooth and topological slice genus Definition
- Worked Seifert-algorithm counts for the trefoil, figure-eight, and kinked unknot Example
- An Alexander-one knot has a trivial Alexander basis Lemma
- Boundary linking is intersection of bounding chains in the four-ball Lemma
- Every integral Seifert form is realized by a knot Lemma
- Homology splitting for the cyclic cover of zero surgery Lemma
- Nonemptiness and minima for the smooth and locally flat slice genera Lemma
- The positive-push Seifert pairing is well-defined for a link surface Lemma
- The skew part of a Seifert matrix is the surface intersection form Lemma
- Well-definedness of the Seifert pairing Lemma
- Cyclic-cover homology order from the Alexander polynomial Theorem
- Knot determinant is the order of double-cover homology Theorem
- The algebraic concordance map is a surjective homomorphism Theorem
Dependency tree · two levels
23 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
- Danny Calegari, Introduction to Knot Theory, §§3.1–3.2, pp.21–26 (standard reference, not scraped)
- Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.3, pp.21–23 (standard reference, not scraped)