The invariant τ of the torus knot grids

On the standard grid torusLink p q of the (p + 1, q + 1) torus link, the X-marking state is a cycle of maximal Alexander grading p q / 2 that generates a tower of the unblocked grid homology GH⁻ (TorusLink/Homology.lean). When p + 1 and q + 1 are coprime the grid is a knot grid, and the invariant τ of the diagram is minus that grading (GridDiagram.two_mul_tau_torusLink): 2 τ = -p q. The sign records that the tower of this grid sits in the top Alexander degree; the book's value τ(T_{p,q}) = (p - 1)(q - 1) / 2 is for the positive torus knot, and τ changes sign under taking the mirror image; the knot this grid presents is the mirror image of that one.

The two smallest knot instances are the 5 × 5 trefoil grid, with τ = -1 (GridDiagram.tau_torusLink_one_two), and the 7 × 7 grid of the (3, 4) torus knot, with τ = -3 (GridDiagram.tau_torusLink_two_three). The unknot grids are the torus link grids with q = 0, so every unknot grid has τ = 0 (GridDiagram.tau_unknot).

These are statements about the diagrams: reading them as values of the concordance invariant of the presented knots requires the invariance of GH⁻ under grid moves, which is not established here.

Main results

• TauCeti.GridDiagram.two_mul_tau_torusLink: 2 τ(torusLink p q) = -p q on a torus knot grid.
• TauCeti.GridDiagram.tau_torusLink_one_two, TauCeti.GridDiagram.tau_torusLink_two_three: τ = -1 on the 5 × 5 trefoil grid and τ = -3 on the 7 × 7 grid of the (3, 4) torus knot.
• TauCeti.GridDiagram.tau_unknot: τ = 0 on every unknot grid.

References

• P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys and Monographs 208, 2015, Chapter 6, where τ of the torus knots is computed from the class of this grid state.

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

τ of a torus knot grid: on the standard grid of the (p + 1, q + 1) torus knot, 2 τ = -p q.

@[simp]

theorem TauCeti.GridDiagram.tau_torusLink_one_two (K : Type u_1) [Field K] [CharP K 2] : ⋯.tau K = -1

On the 5 × 5 trefoil grid, τ = -1.

@[simp]

theorem TauCeti.GridDiagram.tau_torusLink_two_three (K : Type u_1) [Field K] [CharP K 2] : ⋯.tau K = -3

On the 7 × 7 grid of the (3, 4) torus knot, τ = -3.

@[simp]

theorem TauCeti.GridDiagram.tau_unknot {p : ℕ} (K : Type u_1) [Field K] [CharP K 2] (hU : (unknot p).IsKnot) : hU.tau K = 0

τ of an unknot grid is zero: the unknot grids are the torus link grids with q = 0.