The non-torsion class of a torus knot grid

The X-marking state of the standard torus link grid torusLink p q is a cycle of the unblocked grid complex GC⁻, and every grid state has an even number of counted rectangles into it (even_card_unblockedRectangles_torusLink_X). By the parity criterion of Homology/NonTorsion.lean, no monomial in the grid variables kills its class XMarkingClass in GH⁻ (X_pow_smul_XMarkingClass_torusLink_ne_zero), for every torus link grid.

When p + 1 and q + 1 are coprime the diagram is a knot grid, GH⁻ is a module over K[U], and the class is homogeneous of Alexander degree p q / 2, the largest Alexander grading of any grid state (alexanderTwoℤ_le_torusLink_X). Hence GH⁻ of a torus knot grid is not a torsion module (not_isTorsion_unblockedHomology_torusLink): it contains a free K[U]-submodule of rank one, the tower generated by the class of the X-marking state, and the largest Alexander degree of a homogeneous non-torsion class is p q / 2 (two_mul_supNonTorsionDegree_torusLink). This is the diagram-level input to the concordance invariant τ of Ozsváth--Stipsicz--Szabó, minus the maximal Alexander grading of a homogeneous non-torsion element of GH⁻; neither τ nor any statement about the knot presented by the grid is made here. The two smallest knot instances are the 5 × 5 trefoil grid, with maximal non-torsion degree 1 (supNonTorsionDegree_torusLink_one_two), and the 7 × 7 grid of the (3, 4) torus knot, with maximal non-torsion degree 3 (supNonTorsionDegree_torusLink_two_three).

The results here are statements about the grid diagram. Reading them as invariants of the knot it presents requires the stabilization and commutation invariance of GH⁻, which is not established here.

Main results

• TauCeti.GridDiagram.X_pow_smul_XMarkingClass_torusLink_ne_zero: no power of a grid variable kills the class of the X-marking state of a torus link grid.
• TauCeti.GridDiagram.alexanderℤ_X_mem_nonTorsionDegrees_torusLink and TauCeti.GridDiagram.not_isTorsion_unblockedHomology_torusLink: on a torus knot grid the class is a homogeneous non-torsion element, so GH⁻ is not torsion.
• TauCeti.GridDiagram.two_mul_supNonTorsionDegree_torusLink: the maximal non-torsion Alexander degree of GH⁻ of a torus knot grid is p q / 2.

References

The computation follows Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Chapter 6, where the class of this state is shown to generate the tower of GH⁻ of a torus knot; the parity argument replaces the appeal to the structure of GH⁻ made there.

theorem TauCeti.GridDiagram.X_pow_smul_XMarkingClass_torusLink_ne_zero (p q : ℕ) (R : Type u_1) [CommRing R] [CharP R 2] (i : Fin (p + 1 + (q + 1))) (k : ℕ) : MvPolynomial.X i ^ k • (torusLink p q).XMarkingClass R ≠ 0

No power of a grid variable kills the class of the X-marking state in the unblocked grid homology of a torus link grid.

theorem TauCeti.GridDiagram.alexanderℤ_X_mem_nonTorsionDegrees_torusLink {p q : ℕ} (h : (p + 1).Coprime (q + 1)) (K : Type u_1) [Field K] [CharP K 2] : ⋯.toOddComponentGridDiagram.alexanderℤ (torusLink p q).X ∈ (⋯.alexanderUnblockedHomologyGrading K).nonTorsionDegrees

On a torus knot grid the Alexander grading p q / 2 of the X-marking state is a non-torsion degree of the K[U]-module GH⁻: the class of that state is a homogeneous element that no power of U kills.

theorem TauCeti.GridDiagram.not_isTorsion_unblockedHomology_torusLink {p q : ℕ} (h : (p + 1).Coprime (q + 1)) (K : Type u_1) [Field K] [CharP K 2] : ¬Module.IsTorsion (Polynomial K) ↑((torusLink p q).unblockedHomology K)

The unblocked grid homology of a torus knot grid is not torsion: it contains the tower generated by the class of the X-marking state.

theorem TauCeti.GridDiagram.two_mul_supNonTorsionDegree_torusLink {p q : ℕ} (h : (p + 1).Coprime (q + 1)) (K : Type u_1) [Field K] [CharP K 2] : 2 * (⋯.alexanderUnblockedHomologyGrading K).supNonTorsionDegree = ↑p * ↑q

The maximal non-torsion Alexander degree of a torus knot grid is p q / 2: twice the supremum of the Alexander degrees of homogeneous non-torsion classes of GH⁻ is p q.

theorem TauCeti.GridDiagram.supNonTorsionDegree_torusLink_one_two (K : Type u_1) [Field K] [CharP K 2] : (⋯.alexanderUnblockedHomologyGrading K).supNonTorsionDegree = 1

On the 5 × 5 trefoil grid the maximal non-torsion Alexander degree of GH⁻ is 1.

theorem TauCeti.GridDiagram.supNonTorsionDegree_torusLink_two_three (K : Type u_1) [Field K] [CharP K 2] : (⋯.alexanderUnblockedHomologyGrading K).supNonTorsionDegree = 3

On the 7 × 7 grid of the (3, 4) torus knot the maximal non-torsion Alexander degree of GH⁻ is 3.