The diagonal reflection of torus link grid links

The standard grid of the (p + 1, q + 1) torus link reflects to the standard grid of the (q + 1, p + 1) torus link (GridDiagram.transpose_torusLink), so the diagonal reflection of grid links (GridLink.transpose) exchanges the grid links the two grids present. In particular every standard (1, q + 1) torus link grid presents the unknot.

Main results

• TauCeti.GridLink.toGridLink_transpose_torusLink: the reflection exchanges the grid links of the standard (p + 1, q + 1) and (q + 1, p + 1) torus link grids.
• TauCeti.GridDiagram.toGridLink_torusLink_zero_left: the standard (1, q + 1) torus link grid presents the unknot.

References

• P. Ozsváth, A. Stipsicz, Z. Szabó, Grid Homology for Knots and Links, AMS Mathematical Surveys and Monographs 208, 2015, Chapter 3: grid diagrams, their symmetries, and grid moves.

theorem TauCeti.GridLink.toGridLink_transpose_torusLink (p q : ℕ) : (GridDiagram.torusLink p q).transpose.toGridLink = (GridDiagram.torusLink q p).toGridLink

The diagonal reflection exchanges the grid links of the standard (p + 1, q + 1) and (q + 1, p + 1) torus link grids.

@[simp]

theorem TauCeti.GridDiagram.toGridLink_torusLink_zero_left (q : ℕ) : (torusLink 0 q).toGridLink = GridLink.unknot

The standard grid of the (1, q + 1) torus link presents the unknot: it is the diagonal reflection of the standard (q + 1, 1) torus link grid, which is a standard unknot grid.