Seifert surfaces and the Seifert genus of a knot #
A Seifert surface for a knot K ⊆ S³ is a compact, connected, orientable surface smoothly
embedded in S³ whose boundary is K. Every knot has one (Seifert's algorithm), and the least
genus of a Seifert surface is the genus g(K) of the knot, its Seifert genus. It vanishes
exactly for the unknot, it bounds the slice genus from above, and it is the quantity in Giffen's
conjecture on the 0-surgery S³₀(K) (Kirby's problem 1.41 (B)), answered by Gabai.
As for slice surfaces (TauCeti.SmoothSliceSurface), the knot is a smooth circle embedding
K : S¹ → Sⁿ into the unit sphere of an (n + 1)-dimensional real inner product space E, so
that S³ ⊆ ℝ⁴ is the case n = 3. A Seifert surface for K (TauCeti.SeifertSurface K)
bundles
- a compact, connected, orientable smooth surface
Swith boundary, modelled on the half-planeEuclideanHalfSpace 2(orientability isTauCeti.Orientable); - a smooth embedding
Φ : S → Sⁿof manifolds with boundary carrying the boundary ofSonto the image ofK.
The carrier of the surface is a type with instances; it is bundled only because the Seifert genus
is a minimum over all such surfaces. Since Φ is an embedding, the boundary of S is homeomorphic
to the image of K, a circle, so S has exactly one boundary component. Only the image of K
enters, so the Seifert genus does not depend on the parametrization or orientation of the knot.
The genus of a Seifert surface (TauCeti.SeifertSurface.genus) is read off its rational
singular homology, as for slice surfaces: a compact, connected, orientable surface of genus g
with one boundary circle has H₁(S; ℚ) ≅ ℚ²ᵍ, so its genus is half the dimension of H₁(S; ℚ).
The Seifert genus (TauCeti.seifertGenus K) is the infimum of the genera of the Seifert surfaces of
K, valued in ℕ∞ so that it is ⊤ rather than 0 should K have no Seifert surface at all.
(Seifert's algorithm shows that it is finite for n = 3; that existence theorem is not proved
here.) An ambient diffeomorphism of the sphere carries Seifert surfaces of K to Seifert surfaces
of its image, so the Seifert genus is invariant under ambient diffeomorphisms
(TauCeti.seifertGenus_transDiffeomorph), and in particular under ambient isotopy.
For a great circle ι S¹ and a unit vector v orthogonal to ι ℂ, the hemisphere of the great
two-sphere through ι S¹ and -v is a Seifert disc, the image of the closed unit disc of ℂ
under inverse stereographic projection from v of w ↦ 2 ι w
(LinearIsometry.closedBallHemisphereMap). The disc is contractible, so it has genus 0, and
great circles in spheres of dimension at least two, in particular the unknot, have Seifert genus
0 (TauCeti.seifertGenus_greatCircle, TauCeti.seifertGenus_unknot). The converse, that a knot
of Seifert genus 0 is the unknot, needs the classification of compact surfaces and the
Schoenflies theorem, and is not proved here.
Main definitions #
TauCeti.SeifertSurface K: a smooth Seifert surface forK.TauCeti.SeifertSurface.genus: the genusdim H₁(S; ℚ) / 2of a Seifert surface (TauCeti.surfaceGenus).TauCeti.seifertGenus K: the Seifert genus ofK, inℕ∞.TauCeti.SeifertSurface.congrRange,TauCeti.SeifertSurface.transDiffeomorph: a Seifert surface for a knot with the same image, and for the image of the knot under an ambient diffeomorphism.TauCeti.SeifertSurface.hemisphere: the hemisphere Seifert disc of a great circle.
Main results #
TauCeti.seifertGenus_le_genus,TauCeti.le_seifertGenus_iff: the Seifert genus is the infimum of the genera of the Seifert surfaces.TauCeti.seifertGenus_lt_top_iff,TauCeti.exists_genus_eq_seifertGenus: it is finite exactly when a Seifert surface exists, and is then attained.TauCeti.seifertGenus_congr_range,TauCeti.seifertGenus_rotate,TauCeti.seifertGenus_reverse: the Seifert genus only depends on the image of the knot.TauCeti.seifertGenus_transDiffeomorph: it is invariant under ambient diffeomorphisms.TauCeti.SeifertSurface.genus_hemisphere: the hemisphere disc has genus0.TauCeti.seifertGenus_greatCircle,TauCeti.seifertGenus_unknot: great circles inSⁿ,n ≥ 2, and in particular the unknot, have Seifert genus0.
References #
- H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), 571–592.
- W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 2 (Seifert surfaces and the genus of a knot).
- D. Rolfsen, Knots and Links, Publish or Perish (1976), Chapter 5 (Seifert surfaces).
A Seifert surface for a smooth circle embedding K : S¹ → Sⁿ in the unit sphere of E: a
compact, connected, orientable smooth surface with boundary, together with a smooth embedding into
the sphere, in the sense of manifolds with boundary, which carries the boundary of the surface onto
the image of K.
- carrier : Type
The underlying surface.
- topologicalSpace : TopologicalSpace self.carrier
The topology of the surface.
- chartedSpace : ChartedSpace (EuclideanHalfSpace 2) self.carrier
The charts of the surface, modelled on the half-plane.
- isManifold : IsManifold (modelWithCornersEuclideanHalfSpace 2) (↑⊤) self.carrier
The surface is a smooth manifold with boundary.
- compactSpace : CompactSpace self.carrier
The surface is compact.
- connectedSpace : ConnectedSpace self.carrier
The surface is connected.
- orientable : Orientable (modelWithCornersEuclideanHalfSpace 2) self.carrier
The surface is orientable.
- toFun : self.carrier → ↑(Metric.sphere 0 1)
The embedding of the surface into the sphere.
- isSmoothEmbedding : Manifold.IsSmoothEmbedding (modelWithCornersEuclideanHalfSpace 2) (modelWithCornersSelf ℝ (EuclideanSpace ℝ (Fin n))) (↑⊤) self.toFun
The map is a smooth embedding of manifolds with boundary.
The boundary of the surface is carried onto the image of
K.
Instances For
The genus of a Seifert surface: half the dimension of its first rational homology
(TauCeti.surfaceGenus). A compact, connected, orientable surface of genus g with one boundary
circle has H₁(S; ℚ) ≅ ℚ²ᵍ, so this is the genus of S: since S.toFun is an embedding, the
boundary of S is homeomorphic to the image of K, a single circle.
Instances For
The genus of a Seifert surface is the genus TauCeti.surfaceGenus of its carrier.
A Seifert surface for K is a Seifert surface for every knot K' with the same image.
Instances For
Transporting a Seifert surface to a knot with the same image keeps its genus.
Transport of a Seifert surface for K along a diffeomorphism e of the sphere: the same
surface, embedded by e ∘ S.toFun, is a Seifert surface for the transported knot e ∘ K.
Instances For
Transporting a Seifert surface along an ambient diffeomorphism keeps its genus.
The Seifert genus g(K) of a smooth circle embedding K : S¹ → Sⁿ: the least genus of a
Seifert surface for K, or ⊤ if there is none.
Instances For
The Seifert genus is the infimum of the genera of all Seifert surfaces.
The Seifert genus is at most the genus of any Seifert surface.
A lower bound for the Seifert genus is a lower bound for the genus of every Seifert surface.
The Seifert genus is attained when a Seifert surface exists.
The Seifert genus is finite exactly when a Seifert surface exists.
Two knots with the same image have the same Seifert genus.
The Seifert genus does not depend on the rotation of the parametrization.
The Seifert genus does not depend on the orientation of the knot.
The Seifert genus is invariant under ambient diffeomorphisms of the sphere.
The hemisphere disc of a great circle #
The hemisphere Seifert disc of the great circle ι S¹, for a unit vector v orthogonal
to ι ℂ: the closed unit disc of ℂ, embedded in the sphere as the hemisphere centred at -v of
the great two-sphere through ι S¹ and v, by inverse stereographic projection from v of
w ↦ 2 ι w (LinearIsometry.closedBallHemisphereMap).
Instances For
The hemisphere disc has genus 0: the disc is contractible, so its first homology
vanishes.
Great circles have Seifert genus 0 in spheres of dimension at least two: they bound a
hemisphere disc.
The unknot has Seifert genus 0.