flowPolytopeVertices -- generate the polytope associated to a toric quiver

Synopsis

Usage:

flowPolytopeVertices Q

flowPolytopeVertices(W, Q)
Inputs:
- Q, an instance of the type ToricQuiver,
- W, a list, of integer values specifying a weight on the vertices
Optional inputs:
- Format => a string, default value "SimplifiedBasis", specifying what basis to use. The default value SimplifiedBasis returns the polytope in a basis giving the minimal degree necessary for polytope dimension.
Outputs:
- Depending on the option for Format, this method returns a Matrix or List of Lists giving the coordinates of the vertices defining the flow polytope. If the flow polytope does not exist for the given weight. Then, it returns an empty list.

Description

Associated with every acyclic toric quiver and weight pair is a flow polytope. This polytope can be translated to the kernel of the inc map, so it is full dimensional. By default, the output lists the vertices within this vector space. To obtain the presentation in the original fiber, use Format => "FullBasis".

i1 : flowPolytopeVertices(bipartiteQuiver(2, 3))

o1 = {{-1, 1}, {-1, 0}, {1, -1}, {0, -1}, {1, 0}, {0, 1}}

o1 : List

We can vary the weight of the quiver to define the flow polytope

i2 : flowPolytopeVertices({-3,-3,2,2,2}, bipartiteQuiver(2, 3))

o2 = {{-1, 1}, {-1, 0}, {1, -1}, {0, -1}, {1, 0}, {0, 1}}

o2 : List

The user can also recover the polytope $\Delta(Q,\theta)$ within the inverse of the corresponding inc map, that is within $inc^{-1}(\theta)$.

i3 : flowPolytopeVertices( bipartiteQuiver(2, 3), Format => "FullBasis")

o3 = {{2, 0, 1, 0, 2, 1}, {2, 1, 0, 0, 1, 2}, {0, 2, 1, 2, 0, 1}, {1, 2, 0,
     ------------------------------------------------------------------------
     1, 0, 2}, {0, 1, 2, 2, 1, 0}, {1, 0, 2, 1, 2, 0}}

o3 : List

Ways to use flowPolytopeVertices :

flowPolytopeVertices(List,ToricQuiver)
flowPolytopeVertices(ToricQuiver)

For the programmer

The object flowPolytopeVertices is a method function with options.

flowPolytopeVertices -- generate the polytope associated to a toric quiver

Synopsis

Description

See also

Ways to use flowPolytopeVertices :

For the programmer