The braid relation and the Seifert matrix of a closed braid #

The Seifert matrix TauCeti.BraidWord.seifertMatrix w of a braid word w is read off the canonical Seifert surface of its closure: one disc for each strand position, one half-twisted band for each letter, and a basis of loops (t, t') running up the band at position t and down the next band t' on the same level. This file shows that the braid relation σᵢ^ε σᵢ₊₁^ε σᵢ^ε = σᵢ₊₁^ε σᵢ^ε σᵢ₊₁^ε changes this matrix only by a unimodular congruence and a relabelling of the loops. Together with the exchange of letters on distant levels, which only relabels the loops, this covers every Artin relation, so Artin-equivalent braid words have Seifert matrices with the same Alexander polynomial and the same Tristram–Levine signatures. Free cancellation of a letter against its inverse, the other move relating words of the same braid, changes the size of the matrix and is not treated here.

Write p for the position of the first of the three letters. On the left the letters at p, p + 1, p + 2 lie on the levels i, i + 1, i, on the right on the levels i + 1, i, i + 1, and every other letter is unchanged. A loop of the left word is matched with the loop of the right word obtained by exchanging p + 1 and p + 2 in its first position and p and p + 1 in its second position. So the middle loop (p, p + 2) on level i is matched with the middle loop (p, p + 2) on level i + 1, a loop ending at p or p + 1 with the loop ending at the other one, a loop starting at p + 1 or p + 2 with the loop starting at the other one, and every other loop with itself. In the matching basis the right Seifert matrix is P V Pᵀ for the left one V, where the row of P at a loop c is c plus a multiple t c of the middle loop m: t m = -2, so that m itself goes to -m; for ε = 1, t c = 1 for the loops starting at p + 2 or ending at p + 1; for ε = -1, t c = 1 for the loops ending at p or starting at p + 1; and t c = 0 otherwise. The matrix P has determinant -1.

The entries of the two Seifert matrices are compared directly from the formula TauCeti.BraidWord.seifertMatrix_apply. A loop never runs past the three letters on the level i or i + 1, and the middle loop meets no loop that avoids the three letters, so all the comparisons reduce to linear arithmetic on positions.

Main results #

TauCeti.BraidWord.exists_seifertMatrix_append_cons_cons_cons_braid: the braid relation changes the Seifert matrix by a unimodular congruence followed by a relabelling of the loops.
TauCeti.BraidWord.exists_seifertMatrix_of_artinStep: the same for every Artin relation step.
TauCeti.BraidWord.alexander_seifertMatrix_eq_of_artinEquiv, TauCeti.BraidWord.tristramLevineSignature_seifertMatrix_eq_of_artinEquiv: Artin-equivalent braid words have Seifert matrices with the same Alexander polynomial and the same Tristram–Levine signatures.

References #

W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 2 (Seifert's algorithm) and Chapter 6 (Seifert matrices and their change under isotopy of the surface).
J. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies 82 (1974), Chapter 1 (the Artin presentation of the braid group).

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theorem TauCeti.BraidWord.exists_seifertMatrix_append_cons_cons_cons_braid {n : ℕ} (u v : BraidWord n) {i j : Fin (n - 1)} {ε : ℤ ˣ} (hij : ↑j = ↑i + 1) :

∃ (ι : ↥(u ++ (j, ε) :: (i, ε) :: (j, ε) :: v).seifertLoops ≃ ↥(u ++ (i, ε) :: (j, ε) :: (i, ε) :: v).seifertLoops) (P : Matrix ↥(u ++ (i, ε) :: (j, ε) :: (i, ε) :: v).seifertLoops ↥(u ++ (i, ε) :: (j, ε) :: (i, ε) :: v).seifertLoops ℤ), IsUnit P.det ∧ (u ++ (j, ε) :: (i, ε) :: (j, ε) :: v).seifertMatrix = (P * (u ++ (i, ε) :: (j, ε) :: (i, ε) :: v).seifertMatrix * P.transpose).submatrix ⇑ι ⇑ι

The braid relation is a unimodular change of basis of the Seifert matrix. Replacing the three letters σᵢ^ε σᵢ₊₁^ε σᵢ^ε of a braid word by σᵢ₊₁^ε σᵢ^ε σᵢ₊₁^ε changes the Seifert matrix of the canonical Seifert surface only by a congruence P V Pᵀ with P invertible over ℤ, followed by a relabelling of the loops. For the three letters at the positions p, p + 1, p + 2, the middle loop (p, p + 2) on level i becomes minus the middle loop on level i + 1, and a loop with one end among the three letters keeps its other end, possibly picking up the middle loop.

Artin relations #

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theorem TauCeti.BraidWord.exists_seifertMatrix_of_artinStep {n : ℕ} {w w' : BraidWord n} (h : w.ArtinStep w') :

∃ (ι : ↥w'.seifertLoops ≃ ↥w.seifertLoops) (P : Matrix ↥w.seifertLoops ↥w.seifertLoops ℤ), IsUnit P.det ∧ w'.seifertMatrix = (P * w.seifertMatrix * P.transpose).submatrix ⇑ι ⇑ι

An Artin relation step is a unimodular change of basis of the Seifert matrix. Exchanging two letters on distant levels only relabels the loops of the canonical Seifert surface, and the braid relation changes the Seifert matrix by a congruence P V Pᵀ with P invertible over ℤ, followed by a relabelling of the loops.

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