Free cancellation and the Seifert matrix of a closed braid

Inserting an inverse pair σᵢ^ε σᵢ^(-ε) into a braid word adds two bands of opposite twist between the discs Dᵢ and Dᵢ₊₁ of the canonical Seifert surface of the closure, the surface whose Seifert matrix is TauCeti.BraidWord.seifertMatrix. When level i already carries a band, the two new bands form a tube on the old surface, so the Seifert matrix changes by an elementary enlargement up to unimodular congruence: a row enlargement for σᵢ σᵢ⁻¹ and a column enlargement for σᵢ⁻¹ σᵢ. Hence the Alexander polynomial and the Tristram–Levine signatures of the Seifert matrix do not change under a free insertion or deletion that keeps the set of levels carrying a letter. When level i carries no letter, the two new bands join two components of the surface, and the Seifert matrix only gains a zero row and column, so the Tristram–Levine signatures do not change under any free insertion or deletion.

The comparison is made in explicit bases. Write p for the insertion point, so that the new bands sit at positions p and p + 1. The loops after the insertion are the old loops with their bands moved past the insertion point, the loop β = (p, p + 1) through the two new bands, which is the meridian of the tube, and the loops ending at p or starting at p + 1, which exist when a band of level i precedes or follows the insertion point. If both exist, they replace the old loop γ of level i across the insertion point, and γ is the sum of them and β. In the basis where that sum stands in for γ, the old loops keep their entries, the column of β vanishes and its row is ±1 times the coordinate vector of one remaining new loop (for ε = 1; transposed for ε = -1). That is the shape recognised by TauCeti.KnotTheory.exists_mul_mul_transpose_eq_enlargeRow.

For the Alexander polynomial the hypothesis that level i is already in use is needed. On two strands, the empty word has the 0 × 0 Seifert matrix while σ₁ σ₁⁻¹ has the matrix [0], because the canonical surface of the empty word is disconnected.

Main results

• TauCeti.BraidWord.exists_seifertMatrix_insert_congr_enlargeRow: inserting σᵢ σᵢ⁻¹ on a level in use turns the Seifert matrix into a row enlargement of the old one, up to relabelling and a unimodular congruence.
• TauCeti.BraidWord.exists_seifertMatrix_insert_congr_enlargeColumn: the same for σᵢ⁻¹ σᵢ, with a column enlargement.
• TauCeti.BraidWord.exists_seifertMatrix_insert_eq_fromBlocks: inserting an inverse pair on an unused level adds a zero row and column to the Seifert matrix, up to relabelling.
• TauCeti.BraidWord.alexander_seifertMatrix_insert: an insertion on a level in use keeps the Alexander polynomial of the Seifert matrix.
• TauCeti.BraidWord.tristramLevineSignature_seifertMatrix_insert: every insertion keeps the Tristram–Levine signatures of the Seifert matrix.
• TauCeti.BraidWord.FreeCancelStep.alexander_seifertMatrix_eq: a free insertion or deletion step that keeps the set of levels in use keeps the Alexander polynomial.
• TauCeti.BraidWord.FreeCancelStep.tristramLevineSignature_seifertMatrix_eq: every free insertion or deletion step keeps the Tristram–Levine signatures.

References

• W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 6 (Seifert matrices) and Chapter 8 (S-equivalence: adding a tube to a Seifert surface enlarges its Seifert matrix).

Positions

Loops of the canonical Seifert surface after an insertion

Entries of the Seifert matrix after an insertion

The loop γ = (t₀, t₁) of level i running across the insertion point is cut by the two new bands into the loops α = (t₀, p), (p, p + 1) and δ = (p + 1, t₁) (positions after the insertion). As a class on the new surface, γ is their sum, and the entries below say that this sum links every moved loop as γ did.

Matching the loops before and after the insertion

Assembling the congruence

Inserting on an unused level

If no letter of u ++ v lies on level i, the discs Dᵢ and Dᵢ₊₁ lie on different components of the canonical surface, and the two inserted bands join these components. The only new loop is β = (p, p + 1) through the two new bands; it meets no other loop, and its two bands have opposite twists, so the Seifert matrix gains one zero row and column.

theorem TauCeti.BraidWord.exists_seifertMatrix_insert_eq_fromBlocks {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (ε : ℤˣ) (h : ¬∃ x ∈ u ++ v, x.1 = i) : ∃ (ι : ↥(u ++ v).seifertLoops ⊕ Unit ≃ ↥(u ++ (i, ε) :: (i, -ε) :: v).seifertLoops), (u ++ (i, ε) :: (i, -ε) :: v).seifertMatrix.submatrix ⇑ι ⇑ι = Matrix.fromBlocks (u ++ v).seifertMatrix 0 0 0

Inserting an inverse pair on an unused level adds a zero row and column. If no letter of u ++ v lies on level i, inserting σᵢ^ε σᵢ^(-ε) changes the Seifert matrix of the closure, after relabelling the loops by ι, into the block sum of the Seifert matrix of u ++ v with the 1 × 1 zero matrix.

Free insertion is an S-equivalence

theorem TauCeti.BraidWord.card_seifertLoops_insert {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (ε : ℤˣ) (h : ∃ x ∈ u ++ v, x.1 = i) : Fintype.card ↥(u ++ (i, ε) :: (i, -ε) :: v).seifertLoops = Fintype.card ↥(u ++ v).seifertLoops + 2

Inserting an inverse pair on a level already in use adds exactly two Seifert loops.

theorem TauCeti.BraidWord.exists_seifertMatrix_insert_congr_enlargeRow {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (h : ∃ x ∈ u ++ v, x.1 = i) : ∃ (ι : ↥(u ++ v).seifertLoops ⊕ Fin 2 ≃ ↥(u ++ (i, 1) :: (i, -1) :: v).seifertLoops) (P : Matrix (↥(u ++ v).seifertLoops ⊕ Fin 2) (↥(u ++ v).seifertLoops ⊕ Fin 2) ℤ) (η : ↥(u ++ v).seifertLoops → ℤ), IsUnit P.det ∧ P * (u ++ (i, 1) :: (i, -1) :: v).seifertMatrix.submatrix ⇑ι ⇑ι * P.transpose = KnotTheory.enlargeRow (u ++ v).seifertMatrix η

Inserting σᵢ σᵢ⁻¹ enlarges the Seifert matrix by a row. If the level i already carries a letter of u ++ v, inserting the inverse pair σᵢ σᵢ⁻¹ changes the Seifert matrix of the closure, after relabelling the loops by ι and a unimodular change of basis P, into a row enlargement of the Seifert matrix of u ++ v.

theorem TauCeti.BraidWord.exists_seifertMatrix_insert_congr_enlargeColumn {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (h : ∃ x ∈ u ++ v, x.1 = i) : ∃ (ι : ↥(u ++ v).seifertLoops ⊕ Fin 2 ≃ ↥(u ++ (i, -1) :: (i, 1) :: v).seifertLoops) (P : Matrix (↥(u ++ v).seifertLoops ⊕ Fin 2) (↥(u ++ v).seifertLoops ⊕ Fin 2) ℤ) (ξ : ↥(u ++ v).seifertLoops → ℤ), IsUnit P.det ∧ P * (u ++ (i, -1) :: (i, 1) :: v).seifertMatrix.submatrix ⇑ι ⇑ι * P.transpose = KnotTheory.enlargeColumn (u ++ v).seifertMatrix ξ

Inserting σᵢ⁻¹ σᵢ enlarges the Seifert matrix by a column. If the level i already carries a letter of u ++ v, inserting the inverse pair σᵢ⁻¹ σᵢ changes the Seifert matrix of the closure, after relabelling the loops by ι and a unimodular change of basis P, into a column enlargement of the Seifert matrix of u ++ v.

theorem TauCeti.BraidWord.alexander_seifertMatrix_insert {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (ε : ℤˣ) (h : ∃ x ∈ u ++ v, x.1 = i) : KnotTheory.alexander (u ++ (i, ε) :: (i, -ε) :: v).seifertMatrix = KnotTheory.alexander (u ++ v).seifertMatrix

Inserting an inverse pair of letters on a level already in use keeps the Alexander polynomial of the Seifert matrix.

theorem TauCeti.BraidWord.tristramLevineSignature_seifertMatrix_insert {n : ℕ} (u v : BraidWord n) (i : Fin (n - 1)) (ε : ℤˣ) (ω : ℂ) : KnotTheory.tristramLevineSignature ((u ++ (i, ε) :: (i, -ε) :: v).seifertMatrix.map Int.cast) ω = KnotTheory.tristramLevineSignature ((u ++ v).seifertMatrix.map Int.cast) ω

Inserting an inverse pair of letters keeps the Tristram–Levine signatures of the Seifert matrix. On a level already in use the matrix changes by an elementary enlargement, and on an unused level by a zero row and column; neither changes the signatures.

theorem TauCeti.BraidWord.FreeCancelStep.alexander_seifertMatrix_eq {n : ℕ} {w w' : BraidWord n} (h : w.FreeCancelStep w') (hlevel : ∀ (l : Fin (n - 1)), (∃ x ∈ w, x.1 = l) ↔ ∃ x ∈ w', x.1 = l) : KnotTheory.alexander w.seifertMatrix = KnotTheory.alexander w'.seifertMatrix

A free insertion or deletion of an inverse pair that does not change which levels carry a letter keeps the Alexander polynomial of the Seifert matrix.

theorem TauCeti.BraidWord.FreeCancelStep.tristramLevineSignature_seifertMatrix_eq {n : ℕ} {w w' : BraidWord n} (h : w.FreeCancelStep w') (ω : ℂ) : KnotTheory.tristramLevineSignature (w.seifertMatrix.map Int.cast) ω = KnotTheory.tristramLevineSignature (w'.seifertMatrix.map Int.cast) ω

A free insertion or deletion of an inverse pair keeps the Tristram–Levine signatures of the Seifert matrix.