Words of the same braid have Seifert matrices with the same signatures

Two braid words represent the same braid exactly when they are connected by Artin relations and free cancellation of inverse pairs. An Artin relation changes the Seifert matrix TauCeti.BraidWord.seifertMatrix of the closure by a congruence (TauCeti.BraidWord.tristramLevineSignature_seifertMatrix_eq_of_artinStep), and a free insertion or deletion by an elementary enlargement or a zero row and column (TauCeti.BraidWord.FreeCancelStep.tristramLevineSignature_seifertMatrix_eq). None of these changes the Tristram–Levine signatures, so they depend only on the braid.

Main results

• TauCeti.BraidWord.WordEquiv.tristramLevineSignature_seifertMatrix_eq and TauCeti.BraidWord.tristramLevineSignature_seifertMatrix_eq_of_toBraid_eq: words representing the same braid have Seifert matrices with the same Tristram–Levine signatures.

References

• W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 6 (Seifert matrices) and Chapter 8 (the signature).

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

Braid words connected by Artin relations and free cancellation have Seifert matrices with the same Tristram–Levine signatures.

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

The signatures of the Seifert matrix depend only on the braid. Two braid words representing the same braid have Seifert matrices with the same Tristram–Levine signatures.