Documentation

TauCeti.KnotTheory.Signature

The signature of a Seifert matrix #

The signature of a knot is the signature of the symmetrised Seifert matrix V + Vᵀ of any Seifert surface for it. Because Matrix.signature reads only the quadratic form x ↦ x ⬝ᵥ V *ᵥ x, which sees V only through V + Vᵀ, that invariant is Matrix.signature V on the nose (Matrix.signature_add_transpose), and no separate definition is introduced here.

What makes it an invariant of the knot rather than of the chosen surface is that it is unchanged by the two moves generating S-equivalence of Seifert matrices: congruence V ↦ P * V * Pᵀ by a matrix with unit determinant — over ℤ exactly a change of basis of the first homology of the surface — and the two enlargements TauCeti.KnotTheory.enlargeColumn and TauCeti.KnotTheory.enlargeRow that record adding a tube to the surface. Congruence invariance is already Matrix.signature_congr; the enlargements are the content of this file. An enlargement adds a hyperbolic plane, which is what makes it invisible to the signature even though it changes the size of the matrix by two.

The normalisation is pinned by the same two examples as the Alexander polynomial: the trefoil has signature -2 and the figure-eight knot signature 0. Together with Matrix.signature_of_isEmpty for the unknot, these show the invariant is not vacuous, and already distinguish the trefoil from the unknot and from the figure-eight knot. The mirror image, whose Seifert matrix is -Vᵀ, negates the signature.

This is the classical (Murasugi) signature, that is the Tristram--Levine signature at ω = -1; the whole Tristram--Levine family lives over ℂ and needs the Hermitian form (1 - ω) V + (1 - star ω) Vᵀ, which is not built here.

The block identities behind the enlargements hold over an arbitrary commutative ring, and the signature statements over an arbitrary linearly ordered field; the knot-theoretic case is a Seifert matrix over ℤ read in ℚ or ℝ, and the two example theorems are stated that way.

Main results #

References #

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theorem TauCeti.KnotTheory.add_transpose_enlargeColumn {R : Type u_1} [CommRing R] {ι : Type u_2} (V : Matrix ι ι R) (ξ : ι → R) :
enlargeColumn V ξ + (enlargeColumn V ξ).transpose = Matrix.fromBlocks (V + V.transpose) (enlargeBlock ξ) (enlargeBlock ξ).transpose !![0, 1; 1, 0]

The symmetrisation of a column enlargement, in blocks: the old symmetrised matrix, the enlargement vector, and a hyperbolic plane.

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theorem TauCeti.KnotTheory.add_transpose_enlargeRow {R : Type u_1} [CommRing R] {ι : Type u_2} (V : Matrix ι ι R) (η : ι → R) :
enlargeRow V η + (enlargeRow V η).transpose = enlargeColumn V.transpose η + (enlargeColumn V.transpose η).transpose

A row enlargement and the column enlargement of the transpose have the same symmetrisation.

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@[simp]
theorem TauCeti.KnotTheory.signature_enlargeColumn {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {ι : Type u_2} [Fintype ι] (V : Matrix ι ι 𝕜) (ξ : ι → 𝕜) :
(enlargeColumn V ξ).signature = V.signature

The signature is unchanged by a column enlargement of a Seifert matrix. The enlargement adds a hyperbolic plane, which contributes nothing to the signature.

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@[simp]
theorem TauCeti.KnotTheory.signature_enlargeRow {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {ι : Type u_2} [Fintype ι] (V : Matrix ι ι 𝕜) (η : ι → 𝕜) :
(enlargeRow V η).signature = V.signature

The signature is unchanged by a row enlargement of a Seifert matrix.

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theorem TauCeti.KnotTheory.signature_trefoilSeifertMatrix {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] :
(trefoilSeifertMatrix.map Int.cast).signature = -2

The signature of the trefoil is -2. Its symmetrised Seifert matrix !![-2, 1; 1, -2] is negative definite, being congruent to diagonal ![-2, -3/2].

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theorem TauCeti.KnotTheory.signature_figureEightSeifertMatrix {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] :
(figureEightSeifertMatrix.map Int.cast).signature = 0

The signature of the figure-eight knot is 0. Its symmetrised Seifert matrix !![2, 1; 1, -2] is indefinite, being congruent to diagonal ![2, -5/2].

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theorem TauCeti.KnotTheory.signature_trefoil_ne_signature_figureEight {𝕜 : Type u_1} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] :
(trefoilSeifertMatrix.map Int.cast).signature ≠ (figureEightSeifertMatrix.map Int.cast).signature

The trefoil and the figure-eight knot have different signatures, so their Seifert matrices are not S-equivalent.