Ranks of the quantum pieces of circle tensors

The degree-p piece of the circle tensor V^{⊗ ι} of a finite circle set ι is free, with basis the labellings of degree p (TauCeti.Khovanov.circlePieceBasis). This file computes its rank, both as a count of labellings and as the coefficient of q ^ p in (q + q⁻¹) ^ |ι|, the graded rank of V^{⊗ ι}.

For an oriented PD-code D, the chain group of the Khovanov complex in bidegree (i, j) is the direct sum, over the states s of cochain degree i, of the degree-j pieces of the shifted circle tensors V(Circle s)⟨|S| + n₊ - 2 n₋⟩, that is, of the degree j - (|S| + n₊ - 2 n₋) pieces of V(Circle s). The graded Euler characteristic of these chain groups is therefore the signed sum of the ranks of those pieces. TauCeti.OrientedPDCode.coeff_enhancedStateSum shows that this signed sum is the coefficient of q ^ j in the enhanced state sum, which for a nonempty diagram is (q + q⁻¹) times the Jones polynomial at t ^ (1/2) = -q (TauCeti.OrientedPDCode.enhancedStateSum_eq_mul_eval₂_jonesPolynomial). This is the chain-level half of the identification of the graded Euler characteristic of Khovanov homology with the Jones polynomial; Euler–Poincaré for the Khovanov complex supplies the other half.

The ranks are computed over any nontrivial commutative ring, so they apply to the integral complex and to each of its scalar extensions.

Main results

• TauCeti.Khovanov.finrank_circleQuantumGrading_piece: the degree-p piece of the circle tensor has rank the number of labellings of degree p.
• TauCeti.Khovanov.coeff_pow_eq_finrank_circleQuantumGrading_piece: that rank is the coefficient of q ^ p in (q + q⁻¹) ^ |ι|.
• TauCeti.OrientedPDCode.coeff_enhancedStateSum: the coefficient of q ^ j in the enhanced state sum is ∑ₛ (-1) ^ (|S| - n₋) rank V(Circle s)_{j - (|S| + n₊ - 2 n₋)}.

References

• M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), §§2–3.
• D. Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), §1.3 and §2.

theorem TauCeti.Khovanov.finrank_circleQuantumGrading_piece (R : Type u) [CommRing R] [Nontrivial R] {ι : Type u_1} [Fintype ι] [DecidableEq ι] (p : ℤ) : Module.finrank R ↥((circleQuantumGrading R ι).piece ℤ p) = {l : ι → Bool | labelingQuantumDegree ι l = p}.card

The degree-p piece of the circle tensor has rank the number of labellings of degree p.

theorem TauCeti.Khovanov.coeff_pow_eq_finrank_circleQuantumGrading_piece (R : Type u) [CommRing R] [Nontrivial R] {ι : Type u_1} [Fintype ι] (p : ℤ) : ((LaurentPolynomial.T 1 + LaurentPolynomial.T (-1)) ^ Fintype.card ι).coeff p = ↑(Module.finrank R ↥((circleQuantumGrading R ι).piece ℤ p))

The graded rank of the circle tensor: the rank of its degree-p piece is the coefficient of q ^ p in (q + q⁻¹) ^ |ι|.

theorem TauCeti.OrientedPDCode.coeff_enhancedStateSum {n : ℕ} (D : OrientedPDCode n) (R : Type u_1) [CommRing R] [Nontrivial R] (j : ℤ) : D.enhancedStateSum.coeff j = ∑ s : Fin n → Bool, (D.stateCochainDegree s).negOnePow • ↑(Module.finrank R ↥((Khovanov.circleQuantumGrading R (D.ResolvedCircle s)).piece ℤ (j - D.stateQuantumShift s)))

The enhanced state sum as a graded Euler characteristic of circle tensors: the coefficient of q ^ j in the enhanced state sum is the signed sum, over the states s, of the ranks of the degree-j pieces of the shifted circle tensors V(Circle s)⟨|S| + n₊ - 2 n₋⟩, with sign (-1) ^ (|S| - n₋). Here S is the set of crossings smoothed by B, and the ranks may be taken over any nontrivial commutative ring R.