The topological slice genus of a knot #
The topological slice genus g_s^top(K) of a knot K ⊆ S³ is the least genus of a compact,
connected, orientable surface locally flatly and properly embedded in the four-ball D⁴ with
boundary K. It is the topological counterpart of the smooth slice genus TauCeti.sliceGenus,
exactly as topological sliceness (TauCeti.IsTopologicallySlice) is the counterpart of smooth
sliceness: it is 0 on topologically slice knots and at most the smooth slice genus, and the gap
between the two genera is the subject of Freedman's work on topological four-manifolds.
As in TauCeti.KnotTheory.Slice.Genus, the knot is a smooth circle embedding K : S¹ → Sⁿ into
the unit sphere of an (n + 1)-dimensional real inner product space E, with S³ ⊆ ℝ⁴ the case
n = 3. A topological slice surface for K (TauCeti.TopologicalSliceSurface K) bundles
- a compact, connected, orientable surface
Swith boundary, modelled on the half-planeEuclideanHalfSpace 2; - a locally flat embedding
Φ : S → Dⁿ⁺¹(TauCeti.IsLocallyFlat), with tangential model the half-plane and complementary modelℝⁿ⁻¹, which is properly embedded (the preimage of the boundary sphere is the boundary ofS) and carries the boundary ofSonto the image ofK.
Only the embedding is weakened from the smooth case. The abstract surface S still carries a
smooth structure, which is used to state its orientability (TauCeti.Orientable). This is no
restriction: every compact topological surface is homeomorphic to a smooth one (by Radó's
triangulation theorem and the classification of surfaces), and the conditions on Φ and the
genus only see the topology of S. The same convention is used for locally flat slice discs
(TauCeti.IsTopologicalSliceDisc), whose carrier is the smooth closed unit disc.
A topological slice surface extends the slice surface data TauCeti.SliceSurfaceData K shared
with smooth slice surfaces by the local flatness condition. Its genus
(TauCeti.SliceSurfaceData.genus) is thus half the dimension of its first rational homology, as for
smooth slice surfaces, and the topological slice genus (TauCeti.topologicalSliceGenus K) is the
infimum of these genera in ℕ∞.
A smooth slice surface is a neat smooth embedding of manifolds with boundary, hence locally flat
(TauCeti.IsLocallyFlat.of_isSmoothEmbedding_of_preimage_boundary_of_sub_eq), so it is a
topological slice surface of the same genus
(TauCeti.SmoothSliceSurface.toTopologicalSliceSurface). Hence the topological slice genus is at
most the smooth one (TauCeti.topologicalSliceGenus_le_sliceGenus). A locally flat slice disc is
a topological slice surface of genus 0, so topologically slice knots have topological slice genus
0. The converse, that a slice surface of genus 0 is a disc, needs the classification of compact
surfaces and is not proved here.
Main definitions #
TauCeti.TopologicalSliceSurface K: a locally flat slice surface forK.TauCeti.topologicalSliceGenus K: the topological slice genus ofK, inℕ∞.TauCeti.SmoothSliceSurface.toTopologicalSliceSurface: a smooth slice surface, read as a topological one.TauCeti.IsTopologicalSliceDisc.toTopologicalSliceSurface: a locally flat slice disc as a topological slice surface.
Main results #
TauCeti.topologicalSliceGenus_le_genus,TauCeti.le_topologicalSliceGenus_iff: the topological slice genus is the infimum of the genera of the topological slice surfaces.TauCeti.topologicalSliceGenus_lt_top_iff,TauCeti.exists_genus_eq_topologicalSliceGenus: it is finite exactly when a topological slice surface exists, and is then attained.TauCeti.topologicalSliceGenus_le_sliceGenus: the topological slice genus is at most the smooth slice genus.TauCeti.topologicalSliceGenus_congr_range,TauCeti.topologicalSliceGenus_rotate,TauCeti.topologicalSliceGenus_reverse,TauCeti.topologicalSliceGenus_congr_linearIsometryEquiv: it only depends on the image of the knot, and is invariant under linear isometries of the ambient spaces.TauCeti.IsTopologicallySlice.topologicalSliceGenus_eq_zero,TauCeti.topologicalSliceGenus_unknot: topologically slice knots, and in particular the unknot, have topological slice genus0.
References #
- C. Livingston, A survey of classical knot concordance, in Handbook of Knot Theory (2005), Section 2, for the smooth and topological four-ball genera.
- M. Freedman and F. Quinn, Topology of 4-Manifolds, Princeton (1990), Section 11.7, for locally flat surfaces in the four-ball.
A topological slice surface for a smooth circle embedding K : S¹ → Sⁿ in the unit sphere
of E: a compact, connected, orientable surface with boundary, together with a locally flat
embedding into the closed unit ball of E, with tangential model the half-plane and complementary
model ℝⁿ⁻¹, which is properly embedded (the preimage of the boundary sphere is the boundary of the
surface) and carries the boundary of the surface onto the image of K. The surface carries a smooth
structure only to state its orientability; the embedding is merely locally flat. Its genus is the
genus TauCeti.SliceSurfaceData.genus of the underlying slice surface data.
- chartedSpace : ChartedSpace (EuclideanHalfSpace 2) self.carrier
- isManifold : IsManifold (modelWithCornersEuclideanHalfSpace 2) (↑⊤) self.carrier
- compactSpace : CompactSpace self.carrier
- connectedSpace : ConnectedSpace self.carrier
- toFun : self.carrier → ↑(Metric.closedBall 0 1)
- isEmbedding : Topology.IsEmbedding self.toFun
- preimage_boundary : self.toFun ⁻¹' ModelWithCorners.boundary ↑(Metric.closedBall 0 1) = ModelWithCorners.boundary self.carrier
- image_boundary : Subtype.val ∘ self.toFun '' ModelWithCorners.boundary self.carrier = Subtype.val '' Set.range ⇑K
- isLocallyFlat : IsLocallyFlat (EuclideanHalfSpace 2) (EuclideanSpace ℝ (Fin (n - 1))) self.toFun
The surface is locally flatly embedded, flat up to the boundary sphere.
Instances For
A topological slice surface for K is one for every knot K' with the same image.
Instances For
Transporting a topological slice surface to a knot with the same image keeps its genus.
Transport of a topological slice surface for K along a linear isometry e : E ≃ₗᵢ[ℝ] F of
the ambient spaces: the same surface, embedded by e ∘ S.toFun, is a topological slice surface for
K' = e ∘ K.
Instances For
Transporting a topological slice surface along a linear isometry of the ambient spaces keeps its genus.
A smooth slice surface for K, read as a topological slice surface: a neat smooth embedding of
manifolds with boundary is locally flat with tangential model the half-plane and complementary
model ℝⁿ⁻¹.
Instances For
Reading a smooth slice surface as a topological slice surface keeps its genus.
A locally flat slice disc for K, as a topological slice surface whose carrier is the closed
unit disc: the disc is compact, connected and orientable.
Instances For
A locally flat slice disc has genus 0: the disc is contractible, so its first homology
vanishes.
The topological slice genus g_s^top(K) of a smooth circle embedding K : S¹ → Sⁿ: the
least genus of a topological slice surface for K, or ⊤ if there is none.
Instances For
The topological slice genus is the infimum of the genera of all topological slice surfaces.
The topological slice genus is at most the genus of any topological slice surface.
A lower bound for the topological slice genus is a lower bound for the genus of every topological slice surface.
The topological slice genus is attained when a topological slice surface exists.
The topological slice genus is finite exactly when a topological slice surface exists.
The topological slice genus is at most the smooth slice genus: every smooth slice surface is a topological slice surface of the same genus.
Two knots with the same image have the same topological slice genus.
The topological slice genus does not depend on the rotation of the parametrization.
The topological slice genus does not depend on the orientation of the knot.
The topological slice genus is invariant under linear isometries of the ambient spaces: if
K' = e ∘ K for a linear isometry e : E ≃ₗᵢ[ℝ] F, then K' and K have the same topological
slice genus.
Topologically slice knots have topological slice genus 0.
The unknot has topological slice genus 0.