# mergeOnVertex -- join two quivers together by identifying a vertex from each

## Synopsis

• Usage:
mergeOnVertex (Q1, V1, Q2, V2)
mergeOnVertex (Q1, V1, M2, V2)
mergeOnVertex (M1, V1, Q2, V2)
• Inputs:
• Outputs:

## Description

create a new quiver from joining two toricQuivers together by identifying vertex $V1$ in $Q1$ with vertex $V2$ in $Q2$.

 i1 : mergeOnVertex (bipartiteQuiver (2, 3), 1, bipartiteQuiver (2, 3), 0) o1 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1} } IncidenceMatrix => | -1 -1 -1 0 0 0 0 0 0 0 0 0 | | 1 0 0 1 0 0 0 0 0 0 0 0 | | 0 1 0 0 1 0 0 0 0 0 0 0 | | 0 0 1 0 0 1 0 0 0 0 0 0 | | 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 | | 0 0 0 0 0 0 0 0 0 -1 -1 -1 | | 0 0 0 0 0 0 1 0 0 1 0 0 | | 0 0 0 0 0 0 0 1 0 0 1 0 | | 0 0 0 0 0 0 0 0 1 0 0 1 | Q0 => {0, 1, 2, 3, 4, 5, 6, 7, 8} Q1 => {{0, 1}, {0, 2}, {0, 3}, {4, 1}, {4, 2}, {4, 3}, {4, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}} weights => {-3, 2, 2, 2, -6, -3, 2, 2, 2} o1 : ToricQuiver
 i2 : mergeOnVertex (bipartiteQuiver (2, 3), 1, matrix({{-1,-1,-1,-1},{1,1,0,0},{0,0,1,1}}), 0) o2 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1, 1, 1, 1, 1} } IncidenceMatrix => | -1 -1 -1 0 0 0 0 0 0 0 | | 1 0 0 1 0 0 0 0 0 0 | | 0 1 0 0 1 0 0 0 0 0 | | 0 0 1 0 0 1 0 0 0 0 | | 0 0 0 -1 -1 -1 -1 -1 -1 -1 | | 0 0 0 0 0 0 1 1 0 0 | | 0 0 0 0 0 0 0 0 1 1 | Q0 => {0, 1, 2, 3, 4, 5, 6} Q1 => {{0, 1}, {0, 2}, {0, 3}, {4, 1}, {4, 2}, {4, 3}, {4, 5}, {4, 5}, {4, 6}, {4, 6}} weights => {-3, 2, 2, 2, -7, 2, 2} o2 : ToricQuiver i3 : mergeOnVertex (matrix({{-1,-1,-1,-1},{1,1,0,0},{0,0,1,1}}), 1, bipartiteQuiver (2, 3), 0) o3 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1, 1, 1, 1, 1} } IncidenceMatrix => | -1 -1 -1 -1 0 0 0 0 0 0 | | 0 0 1 1 0 0 0 0 0 0 | | 1 1 0 0 -1 -1 -1 0 0 0 | | 0 0 0 0 0 0 0 -1 -1 -1 | | 0 0 0 0 1 0 0 1 0 0 | | 0 0 0 0 0 1 0 0 1 0 | | 0 0 0 0 0 0 1 0 0 1 | Q0 => {0, 1, 2, 3, 4, 5, 6} Q1 => {{0, 2}, {0, 2}, {0, 1}, {0, 1}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}} weights => {-4, 2, -1, -3, 2, 2, 2} o3 : ToricQuiver

## Ways to use mergeOnVertex :

• "mergeOnVertex(Matrix,ZZ,ToricQuiver,ZZ)"
• "mergeOnVertex(ToricQuiver,ZZ,Matrix,ZZ)"
• "mergeOnVertex(ToricQuiver,ZZ,ToricQuiver,ZZ)"
• mergeOnVertex(Matrix,ZZ,Matrix,ZZ) (missing documentation)

## For the programmer

The object mergeOnVertex is .