Invariance of the Kauffman bracket under the third Reidemeister move

The eight local smoothings are reduced to the five matchings of the six boundary ports. The reduction inserts six internal vertices into the traversal, except for the middle smoothing, which leaves an additional circle. The surrounding diagram is arbitrary. The crossing weights cancel for each of the six acyclic height orders; kauffmanBracket_reidemeisterThree states the resulting invariance.

References

• L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395-407.
• W. B. R. Lickorish, An Introduction to Knot Theory, Springer GTM 175 (1997), Chapter 3.

The eight local smoothings

@[simp]

theorem TauCeti.PDCode.kauffmanBracket_reidemeisterThree {n : ℕ} (D : PDCode n) (c : Fin 3 ↪ Fin n) (h : D.HasReidemeisterThreeTriangle c) {R : Type u_1} [CommRing R] (a : Rˣ) : (D.reidemeisterThree c).kauffmanBracket a = D.kauffmanBracket a

The third Reidemeister move preserves the Kauffman bracket, for every surrounding PD-code and all six acyclic strand height orders.