Markov stabilization of the Jimbo weighted trace

Adding a strand and a positive crossing multiplies the weighted trace by q ^ N; a negative crossing multiplies it by q ^ (-N). These identities hold over any commutative ring with a unit parameter, including at q = 1, without dividing by the quantum dimension or by q - q⁻¹.

The added strand keeps its colour under the included braid. Consequently, in the diagonal coefficient of a stabilized braid, only the colour-preserving part of the last crossing survives. The weighted partial trace of that crossing supplies the stabilization factor. These are the two Markov laws needed for writhe normalization.

References

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

We use the strand-inclusion coefficient formula and the two-colour partial trace from the companion modules StrandInclusion and Enhancement.

@[simp]

theorem TauCeti.KnotTheory.jimboWeightedTrace_stabilize {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b : BraidGroup (n + 1)) : jimboWeightedTrace q (BraidGroup.strandIncl b * BraidGroup.sigma (Fin.last n)) = ↑(q ^ N) * jimboWeightedTrace q b

Positive Markov stabilization multiplies the weighted trace by q ^ N.

@[simp]

theorem TauCeti.KnotTheory.jimboWeightedTrace_stabilizeInv {R : Type u_1} [CommRing R] {N n : ℕ} (q : Rˣ) (b : BraidGroup (n + 1)) : jimboWeightedTrace q (BraidGroup.strandIncl b * (BraidGroup.sigma (Fin.last n))⁻¹) = ↑(q⁻¹ ^ N) * jimboWeightedTrace q b

Negative Markov stabilization multiplies the weighted trace by q ^ (-N).