symmetry#

Symmetry operations on planar knot diagrams.

Functions

`flip`(k[, nodes, inplace])	Flip the diagram by 180° around each selected node.
`mirror`(k[, crossings, inplace])	Mirror a planar diagram.
`reverse`(k[, inplace])	Reverse orientation of an oriented diagram.

mirror(k, crossings=None, inplace=False)#

Mirror a planar diagram.

For each crossing in crossings (or all crossings, if omitted), apply the permutation (1,2,3,0) which performs a quarter-rotation that mirrors the local crossing structure. Works for oriented and unoriented diagrams.

Parameters:

k (PlanarDiagram | OrientedPlanarDiagram) – Diagram to mirror.
crossings (set | list | tuple | None) – Optional container of crossing node labels to mirror. If None, all crossings are mirrored.
inplace (bool) – If True, modify k in place, otherwise return a copy.

Returns:

The mirrored diagram (k if inplace=True; otherwise a new instance).

flip(k, nodes=None, inplace=False)#

Flip the diagram by 180° around each selected node.

This reverses the cyclic order of incident endpoints at the specified nodes. In S³ / ℝ³ this does not change the knot type or planar diagram type.

Parameters:

k (PlanarDiagram | OrientedPlanarDiagram) – Diagram to flip.
nodes (set | list | tuple | None) – Optional container of node labels to flip. If None, all nodes are flipped.
inplace (bool) – If True, modify k in place, otherwise return a copy.

Returns:

The flipped diagram (k if inplace=True; otherwise a new instance).

reverse(k, inplace=False)#

Reverse orientation of an oriented diagram.

Swaps each arc’s endpoint types (ingoing/outgoing) accordingly.

Parameters:

k (OrientedPlanarDiagram) – Oriented planar diagram to reverse.
inplace (bool) – If True, modify k in place, otherwise return a copy.

Returns:

The orientation-reversed diagram.

Raises:

TypeError – If k is an unoriented PlanarDiagram.

Return type:

OrientedPlanarDiagram