Removing a full twist from a torus link grid

Write P = p + 1 and Q = q + 1. As a braid closure, the (P, Q + P) torus link is the (P, Q) torus link with one more full twist Δ² on its P strands. The full twist is the square of the half twist Δ, a positive braid with P (P - 1) / 2 = (P choose 2) crossings, so changing these crossings turns Δ² into Δ⁻¹ Δ, the trivial braid. This file carries out that argument on the standard torus link grids (GridDiagram.torusLink): the grid links of the standard grids of the (P, Q + P) and the (P, Q) torus links are joined by at most P choose 2 crossing changes (GridLink.crossingChangeGraph_edist_torusLink_add_le).

The proof runs through a family of intermediate grids. For m ≤ P, the private grid twistGrid p q m of size P + Q + m has its O-markings on the diagonal and the X-marking of column j in row j + P + Q for j < m, in row j + Q for m ≤ j < P, and in row j - P for j ≥ P. With m = P it is the standard grid of the (P, Q + P) torus link, and with m = 0 the standard grid of the (P, Q) torus link. From m + 1 to m, move the first column, with its markings in the rows 0 and P + Q, to the right, one adjacent column at a time, until it reaches column P - 1.

• The first m exchanges pass the columns j < m + 1, whose markings lie in the rows j and j + P + Q. These segments interleave with that of the moving column, so each exchange is a cross-commutation, a crossing change.
• The remaining exchanges pass the columns m + 1 ≤ j < P, whose segments, from row j to row j + Q, are nested inside that of the moving column, so each exchange is a commutation.
• The moving column now sits just before column P, whose X-marking is in row 0, next to the O-marking of the moving column. The result is the X-stabilization of twistGrid p q m splitting the X-marking of column P - 1, with the new column just before the split column and the new row at the bottom, which is connected to twistGrid p q m by grid moves (GridDiagram.movesTo_stabilizeX_castSucc).

So the m-th step costs m crossing changes, and the P steps together cost 0 + 1 + ⋯ + (P - 1) = P choose 2.

Main results

• TauCeti.GridLink.crossingChangeGraph_edist_torusLink_add_le: the grid links of the standard grids of the (P, Q + P) and the (P, Q) torus links are at distance at most P choose 2 in the crossing change graph.

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, the unknotting number, and the Milnor conjecture u(T_{p,q}) = (p - 1) (q - 1) / 2.

The intermediate grids

Sliding the first column to the right

Crossing-change distances

theorem TauCeti.GridLink.crossingChangeGraph_edist_torusLink_add_le (p q : ℕ) : crossingChangeGraph.edist (GridDiagram.torusLink p (q + (p + 1))).toGridLink (GridDiagram.torusLink p q).toGridLink ≤ ↑((p + 1).choose 2)

Removing a full twist from a torus link costs (p + 1) choose 2 crossing changes. The grid links of the standard grids of the (p + 1, (q + (p + 1)) + 1) and the (p + 1, q + 1) torus links are at distance at most (p + 1) choose 2 in the crossing change graph: the former has one more full twist on its p + 1 strands, and changing the crossings of a half twist removes it.