The Euler characteristic of the unnormalized Khovanov complex

Let D be a planar PD-code with n crossings. In each quantum degree j, the unnormalized Khovanov complex ⟦D⟧ʲ = TauCeti.PDCode.khovanovBracket D hD R j is the signed totalization of the degree-j cube of resolutions, whose vertex S is the degree j - |S| piece of the circle tensor of the resolution of S. These pieces are finite free, so ⟦D⟧ʲ is a complex of finite free modules concentrated in the cochain degrees 0, …, n. By Euler–Poincaré for cubes, over a Noetherian ring with the rank-nullity property (such as ℤ or a field) the Euler characteristic of the cohomology of ⟦D⟧ʲ is ∑ S, (-1) ^ |S| rank V(S)_{j - |S|}.

For an oriented code with n₊ positive and n₋ negative crossings, the normalized Khovanov complex in bidegree (i, j) is ⟦D⟧ in bidegree (i + n₋, j - n₊ + 2 n₋). Comparing with the coefficientwise formula TauCeti.OrientedPDCode.coeff_enhancedStateSum for the enhanced state sum shows that the coefficient of q ^ j of the enhanced state sum is (-1) ^ n₋ times the Euler characteristic of the cohomology of ⟦D⟧ in quantum degree j - n₊ + 2 n₋. For a nonempty diagram the enhanced state sum is (q + q⁻¹) times the Jones polynomial at t ^ (1/2) = -q, so this identifies, quantum degree by quantum degree, the graded Euler characteristic of the cohomology of the unnormalized Khovanov complex, regraded by the normalization shifts, with (q + q⁻¹) times the Jones polynomial.

Main results

• TauCeti.PDCode.eulerChar_khovanovBracket: the Euler characteristic of ⟦D⟧ʲ is ∑ S, (-1) ^ |S| rank V(S)_{j - |S|}.
• TauCeti.PDCode.homologyEulerChar_khovanovBracket: Euler–Poincaré for ⟦D⟧ʲ, the Euler characteristic of its cohomology is the same signed sum.
• TauCeti.OrientedPDCode.coeff_enhancedStateSum_eq_homologyEulerChar_khovanovBracket: the coefficient of q ^ j of the enhanced state sum is (-1) ^ n₋ times the Euler characteristic of the cohomology of ⟦D⟧ in quantum degree j - n₊ + 2 n₋.
• TauCeti.OrientedPDCode.coeff_mul_eval₂_jonesPolynomial_eq_homologyEulerChar_khovanovBracket: for a nonempty diagram, the coefficient of q ^ j of (q + q⁻¹) times the Jones polynomial at t ^ (1/2) = -q is (-1) ^ n₋ times the Euler characteristic of the cohomology of ⟦D⟧ in quantum degree j - 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.

instance TauCeti.PDCode.instFreeCarrierObjFinsetFinModuleCatResolutionCube {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) (S : Finset (Fin n)) : Module.Free R ↑((D.resolutionCube hD R j).obj S)

The vertices of the cube of resolutions are free: they are homogeneous pieces of circle tensors.

instance TauCeti.PDCode.instFiniteCarrierObjFinsetFinModuleCatResolutionCube {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) (S : Finset (Fin n)) : Module.Finite R ↑((D.resolutionCube hD R j).obj S)

The vertices of the cube of resolutions are finitely generated: they are homogeneous pieces of circle tensors.

instance TauCeti.PDCode.instFreeCarrierXIntModuleCatKhovanovBracket {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j r : ℤ) : Module.Free R ↑((D.khovanovBracket hD R j).X r)

The terms of the unnormalized Khovanov complex are free.

instance TauCeti.PDCode.instFiniteCarrierXIntModuleCatKhovanovBracket {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j r : ℤ) : Module.Finite R ↑((D.khovanovBracket hD R j).X r)

The terms of the unnormalized Khovanov complex are finitely generated.

instance TauCeti.PDCode.instIsStrictlyGEModuleCatKhovanovBracketOfNatInt {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) : (D.khovanovBracket hD R j).IsStrictlyGE 0

The unnormalized Khovanov complex vanishes in negative cochain degrees.

instance TauCeti.PDCode.instIsStrictlyLEModuleCatKhovanovBracketCastInt {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) : (D.khovanovBracket hD R j).IsStrictlyLE ↑n

The unnormalized Khovanov complex of a code with n crossings vanishes in cochain degrees above n.

theorem TauCeti.PDCode.eulerChar_khovanovBracket {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) [Nontrivial R] : HomologicalComplex.eulerChar (D.khovanovBracket hD R j) = ∑ S : Finset (Fin n), ↑(↑S.card).negOnePow * ↑(Module.finrank R ↥((Khovanov.circleQuantumGrading R (D.ResolvedCircle (cubeState S))).piece ℤ (j - ↑S.card)))

The Euler characteristic of the unnormalized Khovanov complex. Over a nontrivial commutative ring, the Euler characteristic of the degree-j component of the unnormalized Khovanov complex of a planar code is the sum over the vertices S of the cube of resolutions of the rank of the degree j - |S| piece of the circle tensor of the resolution of S, with sign (-1) ^ |S|.

theorem TauCeti.PDCode.homologyEulerChar_khovanovBracket {n : ℕ} (D : PDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] (j : ℤ) [IsNoetherianRing R] [HasRankNullity.{u, u} R] : HomologicalComplex.homologyEulerChar (D.khovanovBracket hD R j) = ∑ S : Finset (Fin n), ↑(↑S.card).negOnePow * ↑(Module.finrank R ↥((Khovanov.circleQuantumGrading R (D.ResolvedCircle (cubeState S))).piece ℤ (j - ↑S.card)))

Euler–Poincaré for the unnormalized Khovanov complex. Over a Noetherian commutative ring with the rank-nullity property, such as ℤ or a field, the Euler characteristic of the cohomology of the degree-j component of the unnormalized Khovanov complex of a planar code is the sum over the vertices S of the cube of resolutions of the rank of the degree j - |S| piece of the circle tensor of the resolution of S, with sign (-1) ^ |S|. Over ℤ the rank of a cohomology group is the rank of its free part, so torsion does not contribute.

theorem TauCeti.OrientedPDCode.coeff_enhancedStateSum_eq_homologyEulerChar_khovanovBracket {n : ℕ} (D : OrientedPDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] [IsNoetherianRing R] [HasRankNullity.{u, u} R] (j : ℤ) : D.enhancedStateSum.coeff j = ↑(↑D.negativeCrossingCount).negOnePow * HomologicalComplex.homologyEulerChar (D.khovanovBracket hD R (j - (↑D.positiveCrossingCount - 2 * ↑D.negativeCrossingCount)))

The enhanced state sum as the Euler characteristic of Khovanov cohomology. For a planar oriented code with n₊ positive and n₋ negative crossings, the coefficient of q ^ j of the enhanced state sum is (-1) ^ n₋ times the Euler characteristic of the cohomology of the unnormalized Khovanov complex in quantum degree j - n₊ + 2 n₋, computed over any Noetherian commutative ring with the rank-nullity property, such as ℤ or a field.

theorem TauCeti.OrientedPDCode.coeff_mul_eval₂_jonesPolynomial_eq_homologyEulerChar_khovanovBracket {n : ℕ} (D : OrientedPDCode n) (hD : D.IsPlanar) (R : Type u) [CommRing R] [IsNoetherianRing R] [HasRankNullity.{u, u} R] (j : ℤ) (h : 0 < D.componentCount) : ((LaurentPolynomial.T 1 + LaurentPolynomial.T (-1)) * (LaurentPolynomial.eval₂ (Int.castRingHom (LaurentPolynomial ℤ)) (-LaurentPolynomial.genericUnit ℤ)) D.jonesPolynomial).coeff j = ↑(↑D.negativeCrossingCount).negOnePow * HomologicalComplex.homologyEulerChar (D.khovanovBracket hD R (j - (↑D.positiveCrossingCount - 2 * ↑D.negativeCrossingCount)))

The Jones polynomial as the Euler characteristic of Khovanov cohomology. For a nonempty planar oriented code with n₊ positive and n₋ negative crossings, the coefficient of q ^ j of (q + q⁻¹) times the Jones polynomial at t ^ (1/2) = -q is (-1) ^ n₋ times the Euler characteristic of the cohomology of the unnormalized Khovanov complex in quantum degree j - n₊ + 2 n₋, computed over any Noetherian commutative ring with the rank-nullity property, such as ℤ or a field. The hypothesis 0 < componentCount says that the diagram is nonempty.