Unknotting numbers of torus link grids

This file bounds the unknotting number (GridLink.unknottingNumber) of the grid links presented by the standard torus link grids. The Hopf link has unknotting number ⊤, since it has two components. Removing a full twist on P strands costs at most P choose 2 crossing changes (GridLink.crossingChangeGraph_edist_torusLink_add_le), so the (P, k P + 1) torus knot is unknotted by k (P choose 2) = (P - 1) ((k P + 1) - 1) / 2 crossing changes: the upper bound u(T_{P,Q}) ≤ (P - 1) (Q - 1) / 2 of the Milnor conjecture for Q ≡ 1 mod P. This covers the (3, 4) torus knot, unknotted by three crossing changes.

For two strands a full twist is one explicit crossing change in the standard grid of every size. Exchanging the first two columns of the standard grid of the (2, q + 3) torus link is a cross-commutation, and the resulting diagram is obtained from the standard grid of the (2, q + 1) torus link by two X-stabilizations, up to a cyclic permutation of the columns and row commutations. The (2, 2k + 1) torus knot has unknotting number at most k, the case P = 2 of the general bound.

The unknotting number of the standard torus link grid is symmetric in the two winding numbers (GridLink.unknottingNumber_toGridLink_torusLink_comm): the standard grids of the (p + 1, q + 1) and (q + 1, p + 1) torus links are diagonal reflections of each other (GridDiagram.transpose_torusLink), and the reflection of grid links preserves unknotting numbers (GridLink.unknottingNumber_transpose). So every bound proved here also holds with the two winding numbers exchanged. In particular the (2 k + 1, 2) torus knot has unknotting number at most k.

The bound holds for every torus knot (GridLink.unknottingNumber_toGridLink_torusLink_le). Since (P - 1) (Q + P - 1) / 2 = P choose 2 + (P - 1) (Q - 1) / 2, removing full twists and exchanging the winding numbers, following Euclid's algorithm on coprime P and Q, reduces the bound to the (1, 1) torus knot, the unknot.

Main results

• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_one_one: the Hopf link has unknotting number ⊤.
• TauCeti.GridLink.isCrossingChange_toGridLink_torusLink_one_add_two: the (2, q + 3) and (2, q + 1) torus links differ by one crossing change.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_add_le: removing a full twist from a torus link grid lowers the unknotting number by at most P choose 2.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_comm: the unknotting number of the standard (p + 1, q + 1) torus link grid equals that of the (q + 1, p + 1) one.
• TauCeti.GridLink.isCrossingChange_toGridLink_torusLink_add_two_one: the (q + 3, 2) and (q + 1, 2) torus links differ by one crossing change.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_le: the upper bound of the Milnor conjecture, the (P, Q) torus knot has unknotting number at most (P - 1) (Q - 1) / 2.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_mul_le: the (P, k P + 1) torus knot has unknotting number at most k (P choose 2).
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_one_two_mul_le: the (2, 2k + 1) torus knot has unknotting number at most k.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_two_three_le_three: the (3, 4) torus knot has unknotting number at most three.
• TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_two_mul_one_le: the (2 k + 1, 2) torus knot has unknotting number at most k.

References

• P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys and Monographs 208, 2015, Chapter 6: crossing changes as cross-commutations and the unknotting number; Chapter 3: commutation and stabilization moves.
• P. B. Kronheimer, T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology 32 (1993): the Milnor conjecture u(T_{P,Q}) = (P - 1) (Q - 1) / 2, whose upper bound is classical.

A crossing change between the (2, q + 3) and (2, q + 1) torus link grids

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_one_one : (GridDiagram.torusLink 1 1).toGridLink.unknottingNumber = ⊤

The Hopf link, presented by the standard 4 × 4 grid of the (2, 2) torus link, has unknotting number ⊤: it has two components, so no sequence of crossing changes reaches the unknot.

theorem TauCeti.GridLink.isCrossingChange_toGridLink_torusLink_one_add_two (q : ℕ) : (GridDiagram.torusLink 1 (q + 2)).toGridLink.IsCrossingChange (GridDiagram.torusLink 1 q).toGridLink

The (2, q + 3) and (2, q + 1) torus links differ by one crossing change: exchanging the first two columns of the standard grid of the former is a cross-commutation, and the result is a twofold stabilization of the standard grid of the latter, up to commutation and cyclic permutation.

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_add_le (p q : ℕ) : (GridDiagram.torusLink p (q + (p + 1))).toGridLink.unknottingNumber ≤ ↑((p + 1).choose 2) + (GridDiagram.torusLink p q).toGridLink.unknottingNumber

Removing a full twist: the unknotting number of the grid link of the standard (p + 1, (q + (p + 1)) + 1) torus link grid exceeds that of the (p + 1, q + 1) torus link grid by at most (p + 1) choose 2.

The transposed torus link grids

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_comm (p q : ℕ) : (GridDiagram.torusLink q p).toGridLink.unknottingNumber = (GridDiagram.torusLink p q).toGridLink.unknottingNumber

The unknotting number of a torus link grid is symmetric in its two winding numbers. The standard grids of the (p + 1, q + 1) and (q + 1, p + 1) torus links present grid links with the same unknotting number, since they are diagonal reflections of each other.

theorem TauCeti.GridLink.isCrossingChange_toGridLink_torusLink_add_two_one (q : ℕ) : (GridDiagram.torusLink (q + 2) 1).toGridLink.IsCrossingChange (GridDiagram.torusLink q 1).toGridLink

The (q + 3, 2) and (q + 1, 2) torus links differ by one crossing change: reflect the crossing change between the (2, q + 3) and (2, q + 1) torus links in the diagonal.

Every torus knot

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_le (p q : ℕ) (h : (p + 1).Coprime (q + 1)) : (GridDiagram.torusLink p q).toGridLink.unknottingNumber ≤ ↑(p * q / 2)

The upper bound of the Milnor conjecture. If P = p + 1 and Q = q + 1 are coprime, the (P, Q) torus knot, presented by its standard grid, has unknotting number at most (P - 1) (Q - 1) / 2 = p q / 2.

Following Euclid's algorithm, removing a full twist reduces the (P, Q + P) torus knot to the (P, Q) torus knot at the cost of P choose 2 crossing changes, and transposition exchanges the roles of P and Q, until the (1, 1) torus knot, the unknot, is reached.

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_mul_le (p k : ℕ) : (GridDiagram.torusLink p (k * (p + 1))).toGridLink.unknottingNumber ≤ ↑k * ↑((p + 1).choose 2)

The (p + 1, k (p + 1) + 1) torus knot, presented by its standard grid, has unknotting number at most k ((p + 1) choose 2). This is the upper bound u(T_{P,Q}) ≤ (P - 1) (Q - 1) / 2 of the Milnor conjecture for P = p + 1 and Q = k P + 1, the case of unknottingNumber_toGridLink_torusLink_le in which each of the k full twists is removed by P choose 2 crossing changes.

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_one_two_mul_le (k : ℕ) : (GridDiagram.torusLink 1 (2 * k)).toGridLink.unknottingNumber ≤ ↑k

The (2, 2k + 1) torus knot, presented by its standard grid, has unknotting number at most k: this is the upper bound u(T_{2, 2k+1}) ≤ (2 - 1) (2k + 1 - 1) / 2 of the Milnor conjecture, the case P = 2 of unknottingNumber_toGridLink_torusLink_mul_le.

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_two_three_le_three : (GridDiagram.torusLink 2 3).toGridLink.unknottingNumber ≤ 3

The (3, 4) torus knot, presented by its standard 7 × 7 grid, has unknotting number at most three.

This is the upper bound u(T_{3,4}) ≤ (3 - 1) (4 - 1) / 2 of the Milnor conjecture for this torus knot.

theorem TauCeti.GridLink.unknottingNumber_toGridLink_torusLink_two_mul_one_le (k : ℕ) : (GridDiagram.torusLink (2 * k) 1).toGridLink.unknottingNumber ≤ ↑k

The upper bound of the Milnor conjecture for the (2 k + 1, 2) torus knots. The knot presented by the standard grid of the (2 k + 1, 2) torus knot has unknotting number at most k = (2 k + 1 - 1) (2 - 1) / 2, as its reflection, the (2, 2 k + 1) torus knot, does.