# Translation Between Graphs

Splice diagrams arise from resolution graphs by a reduction procedure and conversely resolution graphs arise from splice diagrams by a continued fraction calculation. At present, Magma only incorporates the former calculation. However, when a splice diagram $s$ has been constructed using a curve singularity, the corresponding resolution graph $g$ is calculated and can be recovered using the function `CorrespondingResolutionGraph`. The vertices of $s$ correspond to a subset of those of $g$. The correspondence can be recovered with the function `CorrespondingVertices`.

## Splice Diagrams from Resolution Graphs

By default Magma always makes the reduced splice diagram since otherwise many determinants would be calculated unnecessarily.

The translation of resolution graph $g$ to splice diagram is done in two steps. First the underlying graph of $g$ is reduced by the removal of all vertices of valency $2$ (including arrows in the valency calculation). Then the edge labels are calculated using determinants of subgraphs.

### `SpliceDiagram(g): GrphRes -> GrphSpl`

```magma
L      : RngIntElt                    Default: 0
K      : RngIntElt                    Default: 0
Reduced: RngIntElt                    Default: 1
```

A splice diagram of the resolution graph $g$. All parameters can take the value $0$ or $1$. If `Reduced` is $1$ then the splice diagram will be reduced, otherwise it will be the splice diagram on the underlying graph of $g$.

The parameter `L` refers to the total linking numbers of the vertices of the resulting splice diagram. The parameter `K` refers to the canonical class of the vertices of the resulting splice diagram. Each quantity will be calculated when the corresponding parameter is $1$.

### `SpliceDiagram(g, v): GrphRes, GrphResVert -> GrphSpl`

The splice diagram of the resolution graph $g$ with the condition that the vertex $v$ will not be removed by a reduction.