topology#

Functions

`bridges`(k)	Return the set of bridges (arcs whose removal disconnects the diagram) using a fast face-based test.
`edges`(k, **endpoint_attributes)	Return ordered strands (“edges”) of the diagram.
`is_bridge`(k, arc_or_endpoint)	Return True if the given arc/endpoint is a bridge (a size-1 arc cut-set).
`is_empty_diagram`(k)	Return True if the diagram has no nodes.
`is_kink`(k, endpoint)	Return True if endpoint forms a kink at a crossing (CCW neighbor is itself).
`is_knot`(k)	Return True if all nodes are crossings and the diagram has a single link component.
`is_knotoid`(k)	Return True if the diagram is a (multi)-knotoid (exactly two degree-1 vertices, rest crossings).
`is_leaf`(k, node)	Return True if node is a degree-1 vertex.
`is_link`(k)	Return True if all nodes are crossings (possibly multiple components).
`is_linkoid`(k)	Return True if the diagram is a multi-linkoid (even number of leaf vertices; others crossings).
`is_loop`(k, arc_or_endpoint)	Return True if an arc/endpoint forms a loop (an arc whose ends are on the same node which is a Vertex).
`is_planar_graph`(k)	Return True if all nodes are vertices (no crossings).
`is_unknot`(k)	Return True if the diagram is a single unknot component.
`is_unlink`(k)	Return True if the diagram is empty or all nodes are unknots (isolated looped vertices).
`kink_region_iterator`(k[, of_node])	Yield singleton “regions” (lists with one endpoint) representing kinks at crossings.
`kinks`(k[, crossing])	Return the set of kink endpoints; optionally restrict to a given crossing.
`leafs`(k)	Return the set of degree-1 vertices.
`loops`(k)	Return a list of arcs (each a 2-endpoint container) that are loops.
`number_of_unknots`(k)	Return the number of degree-2 looped vertices (unknots).
`overstrands`(k)	Return a partition of endpoints into sets that belong to the same overstrand.
`path_from_endpoint`(k, endpoint)	Follow a strand starting at endpoint until reaching a vertex (or a cycle closes).

is_unlink(k)#

Return True if the diagram is empty or all nodes are unknots (isolated looped vertices).

Parameters:: k (PlanarDiagram)
Return type:: bool

is_unknot(k)#

Return True if the diagram is a single unknot component.

Parameters:: k (PlanarDiagram)
Return type:: bool

number_of_unknots(k)#

Return the number of degree-2 looped vertices (unknots).

Parameters:: k (PlanarDiagram)
Return type:: int

is_knot(k)#

Return True if all nodes are crossings and the diagram has a single link component.

Parameters:: k (PlanarDiagram)
Return type:: bool

is_link(k)#

Return True if all nodes are crossings (possibly multiple components).

Parameters:: k (PlanarDiagram)
Return type:: bool

is_planar_graph(k)#

Return True if all nodes are vertices (no crossings).

Parameters:: k (PlanarDiagram)
Return type:: bool

is_empty_diagram(k)#

Return True if the diagram has no nodes.

Parameters:: k (PlanarDiagram)
Return type:: bool

is_knotoid(k)#

Return True if the diagram is a (multi)-knotoid (exactly two degree-1 vertices, rest crossings).

Parameters:: k (PlanarDiagram)
Return type:: bool

is_linkoid(k)#

Return True if the diagram is a multi-linkoid (even number of leaf vertices; others crossings).

Parameters:: k (PlanarDiagram)
Return type:: bool

is_leaf(k, node)#

Return True if node is a degree-1 vertex.

Parameters:: k (PlanarDiagram)
Return type:: bool

leafs(k)#

Return the set of degree-1 vertices.

Parameters:: k (PlanarDiagram)
Return type:: set

is_loop(k, arc_or_endpoint)#

Return True if an arc/endpoint forms a loop (an arc whose ends are on the same node which is a Vertex).

Parameters:

k (PlanarDiagram) – Diagram.
arc_or_endpoint – Either a single Endpoint or a 2-endpoint container (arc).

Return type:

bool

Notes

“Kink” is a special loop notion at crossings; see is_kink.

loops(k)#

Return a list of arcs (each a 2-endpoint container) that are loops.

Parameters:: k (PlanarDiagram)
Return type:: list

is_kink(k, endpoint)#

Return True if endpoint forms a kink at a crossing (CCW neighbor is itself).

Parameters:

k (PlanarDiagram)
endpoint (Endpoint)

Return type:

bool

kinks(k, crossing=None)#

Return the set of kink endpoints; optionally restrict to a given crossing.

Parameters:: k (PlanarDiagram)
Return type:: set

kink_region_iterator(k, of_node=None)#

Yield singleton “regions” (lists with one endpoint) representing kinks at crossings.

Parameters:

k (PlanarDiagram) – Diagram.
of_node – If given, only consider kinks attached to this node.

bridges(k)#

Return the set of bridges (arcs whose removal disconnects the diagram) using a fast face-based test.

Note

This uses a face incidence heuristic (fast) which may not be valid for already disjoint diagrams. For a robust (but slower) cut-set test, use _is_arc_cut_set.

Parameters:: k (PlanarDiagram)
Return type:: set

is_bridge(k, arc_or_endpoint)#

Return True if the given arc/endpoint is a bridge (a size-1 arc cut-set).

Parameters:

k (PlanarDiagram) – Diagram.
arc_or_endpoint – Either an Endpoint (we test the arc with its twin) or a 2-endpoint container.

Raises:

TypeError – if input is neither an Endpoint nor a 2-endpoint container.

Return type:

bool

edges(k, **endpoint_attributes)#

Return ordered strands (“edges”) of the diagram.

Each edge is a list of endpoints starting at a vertex (or forming a closed component) and proceeding through crossings as per path_from_endpoint. Endpoints can be filtered by attributes via keyword arguments (e.g., color=”red”).

Parameters:

k (PlanarDiagram) – Diagram.
**endpoint_attributes – Attribute filters that all endpoints in a strand must satisfy.

Returns:

List of strands, each a list of Endpoint.

Return type:

list[list[Endpoint]]

overstrands(k)#

Return a partition of endpoints into sets that belong to the same overstrand.

Overstrand relation:

At each crossing, pair the two over-passing endpoints.
Along arcs, pair twins.

Returns:: A list of sets (each set is an overstrand’s endpoints).
Parameters:: k (PlanarDiagram | OrientedPlanarDiagram)