Dehn fillings with gluings isotopic to linear ones depend only on the slope

A Dehn filling TauCeti.DehnFilling h s Y of a knot exterior E glues the solid torus V = D² × S¹ to E along a homeomorphism g of the boundary tori S¹ × S¹. Two fillings Y, Y' with gluings g, g' are homeomorphic, by the identity on E, as soon as the torus map g'⁻¹ ∘ g extends to a homeomorphism ψ of V: the homeomorphism is E ∪ V ≅ E ∪ V given by the identity and ψ (TauCeti.DehnFilling.homeomorphOfSolidTorus).

A homeomorphism of S¹ × S¹ induced by an invertible integral matrix (Matrix.GeneralLinearGroup.torusHomeomorph) whose matrix fixes the meridian (1, 0) up to sign, (a, b) ↦ (a ^ ε * b ^ k, b ^ δ) with ε, δ = ±1, extends to V as (z, b) ↦ (z * b ^ k, b ^ δ), with z replaced by its complex conjugate when ε = -1 (TauCeti.SolidTorus.exists_homeomorph_boundaryInclusion_torusHomeomorph). Two matrix gluings with the same filling slope differ by a matrix of this form (IsSolidTorusNeighborhood.exists_homeomorph_boundaryInclusion_symm_apply_of_fillingSlope_eq). If two homeomorphisms g, k of the torus are isotopic, then g⁻¹ ∘ k extends over V too, the isotopy being absorbed into a collar inside V (TauCeti.SolidTorus.exists_homeomorph_boundaryInclusion_symm_apply_of_isotopic). Hence:

• Dehn fillings with isotopic gluings are homeomorphic (TauCeti.DehnFilling.exists_homeomorph_of_isotopic);
• Dehn fillings whose gluings are isotopic to gluings given by integral matrices depend, up to homeomorphism, only on the slope (TauCeti.DehnFilling.exists_homeomorph_of_isotopic_torusHomeomorph): writing A, A' for the matrices, g'⁻¹ ∘ g is the composite of g'⁻¹ ∘ A', A'⁻¹ ∘ A and A⁻¹ ∘ g, each of which extends over V. The two matrix gluings have the same slope as g and g', since homotopic gluings have the same slope (TauCeti.IsSolidTorusNeighborhood.fillingSlope_eq_of_homotopic).

This is the classical well-definedness of Dehn filling (Rolfsen, Knots and Links, 9F), except for one input that is not formalized here: that every homeomorphism of the torus is isotopic to one induced by a matrix.

In particular any filling along the meridian whose gluing is isotopic to a matrix gluing is the ambient space, the ∞-surgery (TauCeti.DehnFilling.nonempty_homeomorph_of_fillingSlope_meridian).

Main results

• TauCeti.IsSolidTorusNeighborhood.fillingSlope_torusHomeomorph: the filling slope of the gluing induced by a matrix A is the slope of the (A₀₀, A₁₀) curve.
• TauCeti.IsSolidTorusNeighborhood.fillingSlope_torusHomeomorph_eq_iff: two matrix gluings have the same filling slope exactly when their first columns agree up to sign.
• TauCeti.IsSolidTorusNeighborhood.exists_fillingSlope_torusHomeomorph_eq: every slope is the filling slope of a matrix gluing.
• IsSolidTorusNeighborhood.exists_homeomorph_boundaryInclusion_symm_apply_of_fillingSlope_eq: two matrix gluings with the same slope differ by a homeomorphism extending over the solid torus.
• TauCeti.DehnFilling.exists_homeomorph_of_isotopic: two fillings with isotopic gluings are homeomorphic, by the identity on the exterior.
• TauCeti.DehnFilling.exists_homeomorph_of_isotopic_torusHomeomorph: two fillings along the same slope with gluings isotopic to matrix gluings are homeomorphic, by the identity on the exterior.

References

• D. Rolfsen, Knots and Links, Publish or Perish (1976), Chapter 9, Section 9F (Dehn surgery: the filled manifold depends only on the curve to which the meridian of the filling solid torus is sent, since a homeomorphism of the boundary torus preserving the meridian extends over the solid torus).

The filling slope of a matrix gluing

theorem TauCeti.IsSolidTorusNeighborhood.fillingSlope_torusHomeomorph {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} (h : IsSolidTorusNeighborhood f Φ) (A : GL (Fin 2) ℤ) : h.fillingSlope A.torusHomeomorph = Slope.mk (FramedBoundaryTorus.torusCurveClass h.exteriorFrontierHomeomorph (↑A 0 0) (↑A 1 0)) ⋯

The filling slope of a matrix gluing. Gluing the solid torus by the homeomorphism induced by an invertible integral matrix A sends its meridian to the (A₀₀, A₁₀) curve, so the filling slope is the slope of that curve, of value A₀₀ / A₁₀ in the framing of Φ.

theorem TauCeti.IsSolidTorusNeighborhood.fillingSlope_torusHomeomorph_eq_iff {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} (h : IsSolidTorusNeighborhood f Φ) (A A' : GL (Fin 2) ℤ) : h.fillingSlope A.torusHomeomorph = h.fillingSlope A'.torusHomeomorph ↔ ↑A 0 0 = ↑A' 0 0 ∧ ↑A 1 0 = ↑A' 1 0 ∨ ↑A 0 0 = -↑A' 0 0 ∧ ↑A 1 0 = -↑A' 1 0

Two matrix gluings have the same filling slope exactly when the first columns of their matrices agree up to sign.

theorem TauCeti.IsSolidTorusNeighborhood.exists_fillingSlope_torusHomeomorph_eq {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} (h : IsSolidTorusNeighborhood f Φ) (s : Slope h.framedBoundaryTorus.H) : ∃ (A : GL (Fin 2) ℤ), h.fillingSlope A.torusHomeomorph = s

Every slope is the filling slope of a matrix gluing. A primitive class with coordinates (p, q) in the framing of Φ is the first column of an invertible integral matrix [[p, -b], [q, a]], with a p + b q = 1, and gluing the solid torus by the homeomorphism of S¹ × S¹ that this matrix induces sends its meridian to the class.

theorem TauCeti.IsSolidTorusNeighborhood.exists_homeomorph_boundaryInclusion_symm_apply_of_fillingSlope_eq {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} (h : IsSolidTorusNeighborhood f Φ) {A A' : GL (Fin 2) ℤ} (hslope : h.fillingSlope A.torusHomeomorph = h.fillingSlope A'.torusHomeomorph) : ∃ (ψ : SolidTorus ≃ₜ SolidTorus), ∀ (q : Circle × Circle), ψ (SolidTorus.boundaryInclusion q) = SolidTorus.boundaryInclusion (A'.torusHomeomorph.symm (A.torusHomeomorph q))

Matrix gluings with the same filling slope differ over the solid torus. If the gluings induced by two invertible integral matrices A, A' have the same filling slope, the homeomorphism A'⁻¹ ∘ A of the boundary torus extends over the solid torus: its matrix fixes the meridian (1, 0) up to sign.

Fillings along the same slope

theorem TauCeti.DehnFilling.exists_homeomorph_of_isotopic {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} {h : IsSolidTorusNeighborhood f Φ} {s s' : Slope h.framedBoundaryTorus.H} {Y : Type u_1} {Y' : Type u_2} [TopologicalSpace Y] [TopologicalSpace Y'] (F : DehnFilling h s Y) (F' : DehnFilling h s' Y') (hFF' : Isotopic ↑F.gluing ↑F'.gluing) : ∃ (e : Y ≃ₜ Y'), ∀ (x : ↑knotExterior), e (F.exteriorEmbedding x) = F'.exteriorEmbedding x

Dehn fillings with isotopic gluings are homeomorphic. If the gluings of two Dehn fillings of the same knot exterior are isotopic homeomorphisms of the torus S¹ × S¹, the filled spaces are homeomorphic, by a homeomorphism that is the identity on the exterior. The two fillings then have the same slope (TauCeti.IsSolidTorusNeighborhood.fillingSlope_eq_of_homotopic).

theorem TauCeti.DehnFilling.exists_homeomorph_of_isotopic_torusHomeomorph {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} {h : IsSolidTorusNeighborhood f Φ} {s : Slope h.framedBoundaryTorus.H} {Y : Type u_1} {Y' : Type u_2} [TopologicalSpace Y] [TopologicalSpace Y'] (F : DehnFilling h s Y) (F' : DehnFilling h s Y') {A A' : GL (Fin 2) ℤ} (hA : Isotopic ↑F.gluing ↑A.torusHomeomorph) (hA' : Isotopic ↑F'.gluing ↑A'.torusHomeomorph) : ∃ (e : Y ≃ₜ Y'), ∀ (x : ↑knotExterior), e (F.exteriorEmbedding x) = F'.exteriorEmbedding x

Dehn fillings with gluings isotopic to matrix gluings depend only on the slope. Two Dehn fillings of the same knot exterior along the same slope, whose gluings are isotopic to homeomorphisms induced by invertible integral matrices, are homeomorphic, by a homeomorphism that is the identity on the exterior.

theorem TauCeti.DehnFilling.nonempty_homeomorph_of_fillingSlope_meridian {X : Type} [TopologicalSpace X] [T2Space X] {f : Circle → X} {Φ : SolidTorus → X} {h : IsSolidTorusNeighborhood f Φ} {Y : Type u_1} [TopologicalSpace Y] (F : DehnFilling h h.framedBoundaryTorus.meridian Y) {A : GL (Fin 2) ℤ} (hA : Isotopic ↑F.gluing ↑A.torusHomeomorph) : Nonempty (Y ≃ₜ X)

∞-surgery. A Dehn filling along the meridian whose gluing is isotopic to a homeomorphism induced by an invertible integral matrix is homeomorphic to the ambient space of the knot.