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Seifert surfaces and smooth knot genus
Definition
Assume the Axiom of Countable Choice. For an oriented smooth knot , define
Every element of is the genus of a connected compact oriented surface. The existence theorem makes nonempty, and the well-ordering of gives its least element. The smooth knot genus is . This genus is the minimum over embedded spanning surfaces; it is not the number of Seifert circles or the genus of every surface produced from a diagram.
Depends on
Used by
- Knot Alexander normalization, symmetry, and genus bound Corollary
- A genus-one spanning surface for the unknot is not minimal Counterexample
- Smooth and topological slice genus Definition
- Trefoil and 5₂ knot Floer genus and fibering cases Example
- Worked Seifert-algorithm counts for the trefoil, figure-eight, and kinked unknot Example
- A knot has genus zero exactly when it is the unknot Lemma
- Knot genus is invariant under mirroring and reversal Lemma
- Nonemptiness and minima for the smooth and locally flat slice genera Lemma
- Nontrivial knot exteriors are irreducible and boundary-incompressible Lemma
- The right trefoil has genus one Lemma
- The tau level is independent of the normalized Heegaard model Lemma
- Euler characteristic and genus of a Seifert disk-and-band surface Theorem
- Existence and uniqueness of prime decomposition for oriented knots Theorem
- Grid homology is independent of the grid and computes knot Floer homology Theorem
- Invariance of the multipointed Heegaard Floer complex Theorem
- Knot Floer homology detects the Seifert genus Theorem
- Knot genus is additive under connected sum Theorem
- Torus-knot Alexander polynomial and genus formulas Theorem
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
- Danny Calegari, Introduction to Knot Theory, §3.1, pp.21–22 (standard reference, not scraped)
- Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.3, pp.21–23 (standard reference, not scraped)