The smooth slice genus of a knot #

The slice genus g_s(K) (or four-ball genus) of a knot K ⊆ S³ is the least genus of a compact, connected, orientable surface smoothly and properly embedded in the four-ball D⁴ with boundary K. It is 0 for the smoothly slice knots, which bound a disc (classically exactly for them, a converse not proved here), and it is the quantity bounded below by the concordance invariants |σ(K)| / 2 and |τ(K)|.

As for sliceness (TauCeti.IsSmoothlySlice), 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 smooth slice surface for K (TauCeti.SmoothSliceSurface K) bundles

a compact, connected, orientable smooth surface S with boundary, modelled on the half-plane EuclideanHalfSpace 2 (orientability is TauCeti.Orientable, the existence of a tangent orientation, including at boundary points);
a smooth embedding Φ : S → Dⁿ⁺¹ of manifolds with boundary, properly embedded (the preimage of the boundary sphere is the boundary of S), which carries the boundary of S onto the image of K.

The carrier of the surface is a type with instances, as everywhere in Tau Ceti; it is bundled here only because the slice 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; and S is Hausdorff, being embedded in the ball. Only the image of K enters, so the slice genus does not depend on the parametrization or orientation of the knot.

The surface, its embedding into the ball and the two boundary conditions are the data TauCeti.SliceSurfaceData K shared with the topological slice surfaces of TauCeti.KnotTheory.Slice.TopologicalGenus, which differ only in asking for a locally flat rather than a smooth embedding; a smooth slice surface extends this data by the smooth embedding condition. The genus, the transports along knots with the same image and along linear isometries, and the disc are constructed once on the shared data.

The genus of a slice surface (TauCeti.SliceSurfaceData.genus) is read off its rational singular homology: 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 one-circle hypothesis holds because the shared data asks the map to be a topological embedding. The slice genus (TauCeti.sliceGenus K) is the infimum of the genera of the slice surfaces of K, valued in ℕ∞ so that it is ⊤ rather than 0 should K have no slice surface at all. (Every knot in S³ bounds a Seifert surface, whose interior can be pushed into D⁴, so g_s(K) is finite for n = 3; that existence theorem is not proved here.)

A smooth slice disc is a slice surface whose carrier is the closed unit disc, which is compact, connected and orientable (TauCeti.orientable_closedBall), and contractible, so its first homology vanishes and its genus is 0. Hence smoothly slice knots, in particular the unknot, have 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.SliceSurfaceData K: a compact, connected, orientable surface properly embedded in the ball with boundary onto K, the data shared by smooth and topological slice surfaces.
TauCeti.SliceSurfaceData.genus: the genus dim H₁(S; ℚ) / 2 of a slice surface.
TauCeti.SliceSurfaceData.ofClosedBall: the closed unit disc as slice surface data.
TauCeti.SmoothSliceSurface K: a smooth slice surface for K.
TauCeti.sliceGenus K: the smooth slice genus of K, in ℕ∞.
TauCeti.IsSmoothSliceDisc.toSmoothSliceSurface: a smooth slice disc as a slice surface.
TauCeti.SmoothSliceSurface.mapLinearIsometryEquiv: a slice surface for K, transported along a linear isometry e of the ambient spaces to a slice surface for e ∘ K.

Main results #

TauCeti.sliceGenus_le_genus, TauCeti.le_sliceGenus_iff: the slice genus is the infimum of the genera of the slice surfaces.
TauCeti.sliceGenus_lt_top_iff, TauCeti.exists_genus_eq_sliceGenus: the slice genus is finite exactly when a slice surface exists, and is then attained.
TauCeti.sliceGenus_congr_range, TauCeti.sliceGenus_rotate, TauCeti.sliceGenus_reverse: the slice genus only depends on the image of the knot.
TauCeti.sliceGenus_congr_linearIsometryEquiv: the slice genus is invariant under linear isometries of the ambient spaces.
TauCeti.SliceSurfaceData.genus_ofClosedBall, TauCeti.IsSmoothSliceDisc.genus_toSmoothSliceSurface: a slice disc has genus 0.
TauCeti.IsSmoothlySlice.sliceGenus_eq_zero, TauCeti.sliceGenus_unknot: smoothly slice knots, and in particular the unknot, have slice genus 0.

References #

C. Livingston, A survey of classical knot concordance, in Handbook of Knot Theory (2005), Section 2, for the four-ball genus.
W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 8, for slice knots and surfaces in D⁴.

The smooth slice genus of a knot #

Main definitions #

Main results #

References #

Slice discs #