Writhe normalization of the Jimbo trace

Multiplying the weighted trace of a braid of exponent sum e by (q ^ N) ^ (-e) cancels both Markov stabilization factors. The result is constant on ordinary unframed Markov equivalence, on the existing braid presentation MarkovBraid.

This is the sl_N specialization of the enhanced braid-trace construction of HOMFLY. Our normalization assigns the quantum dimension ∑ a, jimboWeight q a to the one-strand unknot, rather than one. The normalized trace satisfies the oriented HOMFLY skein relation with a = q ^ N and z = q - q⁻¹. No invertibility of that sum is assumed, so the construction works over arbitrary commutative rings and at q = 1. No geometric-to-diagram or Markov classification theorem is asserted here.

References

• V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527–553 (writhe normalization of enhanced braid traces).
• V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335–388.

The normalization and Markov-equivalence induction follow the existing MarkovBraid.jonesTrace construction in TauCeti.KnotTheory.TemperleyLieb, with the Jimbo stabilization factors in place of the Jones factors.

noncomputable def TauCeti.MarkovBraid.jimboTrace {R : Type u_1} [CommRing R] {N : ℕ} (β : MarkovBraid) (q : Rˣ) : R

The writhe-normalized Jimbo trace with N colours. Its one-strand unknot value is the quantum dimension, with no division by that value.

theorem TauCeti.MarkovBraid.jimboTrace_def {R : Type u_1} [CommRing R] {N : ℕ} (β : MarkovBraid) (q : Rˣ) : β.jimboTrace q = ↑((q ^ N) ^ (-Multiplicative.toAdd ((ArtinGroup.exponentSum (CoxeterMatrix.A (β.predStrands + 1 - 1))) β.braid))) * KnotTheory.jimboWeightedTrace q β.braid

The normalized trace is the weighted trace times the inverse writhe factor.

@[simp]

theorem TauCeti.MarkovBraid.jimboTrace_conj {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b c : BraidGroup (n + 1)) : { predStrands := n, braid := c * b * c⁻¹ }.jimboTrace q = { predStrands := n, braid := b }.jimboTrace q

Conjugation preserves the normalized trace.

@[simp]

theorem TauCeti.MarkovBraid.jimboTrace_stabilize {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b : BraidGroup (n + 1)) : { predStrands := n + 1, braid := BraidGroup.strandIncl b * BraidGroup.sigma (Fin.last n) }.jimboTrace q = { predStrands := n, braid := b }.jimboTrace q

Positive stabilization raises writhe by one and preserves the normalized trace.

@[simp]

theorem TauCeti.MarkovBraid.jimboTrace_stabilizeInv {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b : BraidGroup (n + 1)) : { predStrands := n + 1, braid := BraidGroup.strandIncl b * (BraidGroup.sigma (Fin.last n))⁻¹ }.jimboTrace q = { predStrands := n, braid := b }.jimboTrace q

Negative stabilization lowers writhe by one and preserves the normalized trace.

@[simp]

theorem TauCeti.MarkovBraid.jimboTrace_one {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) : { predStrands := n, braid := 1 }.jimboTrace q = (∑ a : Fin N, KnotTheory.jimboWeight q a) ^ (n + 1)

The identity braid on n + 1 strands has normalized trace equal to the (n + 1)-st power of the quantum dimension. In particular the unknot value is that dimension itself.

theorem TauCeti.MarkovBraid.jimboTrace_skein {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b c : BraidGroup (n + 1)) (i : Fin n) : ↑(q ^ N) * { predStrands := n, braid := b * BraidGroup.sigma i * c }.jimboTrace q - ↑(q⁻¹ ^ N) * { predStrands := n, braid := b * (BraidGroup.sigma i)⁻¹ * c }.jimboTrace q = (↑q - ↑q⁻¹) * { predStrands := n, braid := b * c }.jimboTrace q

The writhe-normalized Jimbo trace satisfies the oriented HOMFLY skein relation with a = q ^ N and z = q - q⁻¹, in any braid context.

theorem TauCeti.IsMarkovMove.jimboTrace_eq {R : Type u_1} [CommRing R] {N : ℕ} {β γ : MarkovBraid} (h : IsMarkovMove β γ) (q : Rˣ) : β.jimboTrace q = γ.jimboTrace q

Every generating Markov move preserves the writhe-normalized Jimbo trace.

theorem TauCeti.MarkovEquiv.jimboTrace_eq {R : Type u_1} [CommRing R] {N : ℕ} {β γ : MarkovBraid} (h : MarkovEquiv β γ) (q : Rˣ) : β.jimboTrace q = γ.jimboTrace q

The writhe-normalized Jimbo trace is constant on unframed Markov equivalence.