The Tristram--Levine signature of a Seifert matrix

For a Seifert matrix V and a parameter ω on the unit circle, the Tristram--Levine form is the Hermitian matrix (1 - ω) V + (1 - conj ω) Vᵀ over ℂ, and the Tristram--Levine signature σ_ω(V) is its signature. It is a one-parameter extension of the classical (Murasugi) signature, recovered at ω = -1 (TauCeti.KnotTheory.tristramLevineSignature_neg_one), and the family is not constant: for the trefoil the value changes from -2 at ω = -1 to 0 at the unit-circle point (3 + 4i)/5, both computed below.

The form is Hermitian for every ω : ℂ, so no hypothesis on ω is imposed here. This file constructs an invariant of a chosen Seifert matrix and proves congruence invariance. The enlargement identities are proved in TristramLevine.Enlargement; the passage to knots and knot concordance is separate. Classically, the ordinary signature at unit-circle parameters away from the relevant Alexander-polynomial roots is a knot-concordance invariant; values at roots require an additional convention such as the averaged signature. The value at ω = 1 is 0 for every V.

Congruence V ↦ P * V * Pᵀ by a matrix with unit determinant — over ℤ a change of basis of the first homology of the Seifert surface — leaves σ_ω unchanged (TauCeti.KnotTheory.tristramLevineSignature_congr). Invariance under the two enlargements TauCeti.KnotTheory.enlargeColumn and TauCeti.KnotTheory.enlargeRow is proved in TristramLevine.Enlargement.

Main definitions

• TauCeti.KnotTheory.tristramLevineForm: the Hermitian form (1 - ω) V + (1 - conj ω) Vᵀ.
• TauCeti.KnotTheory.tristramLevineSignature: its signature.

Main results

• TauCeti.KnotTheory.isHermitian_tristramLevineForm: the form is Hermitian.
• TauCeti.KnotTheory.tristramLevineSignature_neg_one: at ω = -1 the classical signature.
• TauCeti.KnotTheory.tristramLevineSignature_one: at ω = 1 the signature vanishes.
• TauCeti.KnotTheory.tristramLevineSignature_congr: invariance under congruence.
• TauCeti.KnotTheory.tristramLevineSignature_conj: ω and conj ω give the same signature.
• TauCeti.KnotTheory.tristramLevineSignature_neg_transpose: the mirror image, whose Seifert matrix is -Vᵀ, has the negated signature.
• TauCeti.KnotTheory.tristramLevineSignature_neg: so does the reverse mirror image, whose Seifert matrix is -V.
• TauCeti.KnotTheory.tristramLevineSignature_fromBlocks_zero: additivity along a block diagonal.
• TauCeti.KnotTheory.exists_tristramLevineSignature_trefoilSeifertMatrix_ne: the trefoil's signature is not constant on the unit circle.

References

• A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251--264.
• J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98--110.
• C. Livingston, A survey of classical knot concordance, in Handbook of Knot Theory (2005), for the conventions used here.
• W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 8.

def TauCeti.KnotTheory.tristramLevineForm {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : Matrix ι ι ℂ

The Tristram--Levine form of a Seifert matrix V at ω, the Hermitian matrix (1 - ω) V + (1 - conj ω) Vᵀ over ℂ.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_apply {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) (i j : ι) : tristramLevineForm V ω i j = (1 - ω) * ↑(V i j) + (1 - (starRingEnd ℂ) ω) * ↑(V j i)

theorem TauCeti.KnotTheory.isHermitian_tristramLevineForm {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : (tristramLevineForm V ω).IsHermitian

The Tristram--Levine form is Hermitian, for every ω: conjugating an entry exchanges the roles of ω and conj ω, which is what transposing V does.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_zero_matrix {ι : Type u_1} (ω : ℂ) : tristramLevineForm 0 ω = 0

The Tristram--Levine form of the zero matrix vanishes.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_one {ι : Type u_1} (V : Matrix ι ι ℝ) : tristramLevineForm V 1 = 0

The Tristram--Levine form vanishes at ω = 1.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_neg_one {ι : Type u_1} (V : Matrix ι ι ℝ) : tristramLevineForm V (-1) = (2 • (V + V.transpose)).map ⇑(algebraMap ℝ ℂ)

At ω = -1 the Tristram--Levine form is twice the symmetrised Seifert matrix.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_conj {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm V ((starRingEnd ℂ) ω) = (tristramLevineForm V ω).map ⇑(starRingEnd ℂ)

Replacing ω by conj ω conjugates the Tristram--Levine form entrywise.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_transpose {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm V.transpose ω = tristramLevineForm V ((starRingEnd ℂ) ω)

Transposing the Seifert matrix conjugates the parameter of its Tristram--Levine form.

theorem TauCeti.KnotTheory.tristramLevineForm_neg_transpose {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm (-V.transpose) ω = -tristramLevineForm V ((starRingEnd ℂ) ω)

The Seifert matrix -Vᵀ of the mirror image has the negated form at the conjugate parameter.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_neg {ι : Type u_1} (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm (-V) ω = -tristramLevineForm V ω

Negating a Seifert matrix negates its Tristram--Levine form.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_submatrix {ι : Type u_1} {κ : Type u_2} (f : κ → ι) (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm (V.submatrix f f) ω = (tristramLevineForm V ω).submatrix f f

Taking the same submatrix in both coordinates commutes with the Tristram--Levine form.

theorem TauCeti.KnotTheory.tristramLevineForm_congr {ι : Type u_1} [Fintype ι] (P V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineForm (P * V * P.transpose) ω = P.map Complex.ofReal * tristramLevineForm V ω * (P.map Complex.ofReal).conjTranspose

Congruence of Seifert matrices becomes *-congruence of Tristram--Levine forms.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineForm_fromBlocks_zero {ι : Type u_1} {κ : Type u_2} (V : Matrix ι ι ℝ) (W : Matrix κ κ ℝ) (ω : ℂ) : tristramLevineForm (Matrix.fromBlocks V 0 0 W) ω = Matrix.fromBlocks (tristramLevineForm V ω) 0 0 (tristramLevineForm W ω)

The Tristram--Levine form of a block-diagonal Seifert matrix, the Seifert matrix of a connected sum, is the block diagonal of the two forms.

noncomputable def TauCeti.KnotTheory.tristramLevineSignature {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : ℤ

The Tristram--Levine signature of a Seifert matrix V at ω: the signature of the Hermitian form (1 - ω) V + (1 - conj ω) Vᵀ.

theorem TauCeti.KnotTheory.tristramLevineSignature_def {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature V ω = ⋯.signature

The Tristram--Levine signature is the Hermitian signature of its defining form.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_one {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) : tristramLevineSignature V 1 = 0

The Tristram--Levine signature vanishes at ω = 1, where the form itself vanishes.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_zero_matrix {ι : Type u_1} [Fintype ι] (ω : ℂ) : tristramLevineSignature 0 ω = 0

The zero matrix has Tristram--Levine signature zero.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_neg_one {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) : tristramLevineSignature V (-1) = V.signature

At ω = -1 the Tristram--Levine signature is the classical signature of the Seifert matrix.

theorem TauCeti.KnotTheory.tristramLevineSignature_congr {ι : Type u_1} [Fintype ι] [DecidableEq ι] {P : Matrix ι ι ℝ} (hP : IsUnit P.det) (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature (P * V * P.transpose) ω = tristramLevineSignature V ω

Congruence invariance of the Tristram--Levine signature. Replacing V by P * V * Pᵀ for a matrix P with unit determinant does not change it; over ℤ this is a change of basis of the first homology of the Seifert surface.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_conj {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature V ((starRingEnd ℂ) ω) = tristramLevineSignature V ω

Conjugate parameters give the same Tristram--Levine signature, so the family is determined by the upper half of the unit circle.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_transpose {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature V.transpose ω = tristramLevineSignature V ω

Transposing a Seifert matrix preserves its Tristram--Levine signature.

theorem TauCeti.KnotTheory.tristramLevineSignature_neg_transpose {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature (-V.transpose) ω = -tristramLevineSignature V ω

The mirror image negates the Tristram--Levine signature. The mirror of a knot has Seifert matrix -Vᵀ.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_neg {ι : Type u_1} [Fintype ι] (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature (-V) ω = -tristramLevineSignature V ω

Negating a Seifert matrix negates its Tristram--Levine signature. The reverse of the mirror image of a knot, its inverse in the concordance group, has Seifert matrix -V.

theorem TauCeti.KnotTheory.det_tristramLevineForm_neg_ne_zero_iff {ι : Type u_1} [Fintype ι] [DecidableEq ι] (V : Matrix ι ι ℝ) (ω : ℂ) : (tristramLevineForm (-V) ω).det ≠ 0 ↔ (tristramLevineForm V ω).det ≠ 0

The singular parameters of -V are those of V.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_fromBlocks_zero {ι : Type u_1} [Fintype ι] {κ : Type u_2} [Fintype κ] (V : Matrix ι ι ℝ) (W : Matrix κ κ ℝ) (ω : ℂ) : tristramLevineSignature (Matrix.fromBlocks V 0 0 W) ω = tristramLevineSignature V ω + tristramLevineSignature W ω

Additivity of the Tristram--Levine signature along a block diagonal, the Seifert matrix of a connected sum.

@[simp]

theorem TauCeti.KnotTheory.tristramLevineSignature_submatrix_equiv_self {ι : Type u_1} [Fintype ι] {κ : Type u_2} [Fintype κ] (e : ι ≃ κ) (V : Matrix ι ι ℝ) (ω : ℂ) : tristramLevineSignature (V.submatrix ⇑e.symm ⇑e.symm) ω = tristramLevineSignature V ω

Reindexing a Seifert matrix along an equivalence does not change its Tristram--Levine signature.

theorem TauCeti.KnotTheory.exists_tristramLevineSignature_trefoilSeifertMatrix_ne : ∃ (ω : ℂ), ‖ω‖ = 1 ∧ tristramLevineSignature (trefoilSeifertMatrix.map Int.cast) ω ≠ tristramLevineSignature (trefoilSeifertMatrix.map Int.cast) (-1)

The Tristram--Levine signature of the trefoil is not constant on the unit circle. The witness is the rational point (3 + 4i)/5, where the symmetrised form becomes indefinite and the signature changes from -2 to 0.