Enhance concepts.md with structural variation details

Author: hyanwong

docs/concepts.md

@@ -27,6 +27,66 @@ arbitrarily complex patterns of genetic inheritance.
 The Genetic Inheritance Graph Library (giglib) is a proof-of-concept implementation of the idea
 behind GIGs, and is inspired by the standard [tskit](https://tskit.dev) ARG library.
 
-:::{todo}
-Fill out more details from the [README.md](https://github.com/hyanwong/GeneticInheritanceGraphLibrary/blob/main/README.md)
-:::
+## Details
+
+In [`tskit`](https://tskit.dev) we use edge annotations to describe which pieces of DNA are inherited in terms of a left and right coordinate.
+In giglib, this is extended to track the L & R in the edge *child*, and the L & R in the edge *parent* separately.
+The left and right values in each case refer to the coordinate system of the child and parent respectively.
+
+```{note}
+For terminological clarity, we switch to using the term interval-edge (`iedge`)
+to refer to what is normally called an `edge` in a *tskit* Tree Sequence.
+Separating child from parent coordinates brings a host of extra complexities,
+and it’s unclear if the efficiency of the tskit approach,
+with its edge indexing etc, will port in any meaningful way to this new structure.
+```
+
+## Structural variation
+
+Below are some examples of how different sorts of structural variation can be encoded. These correspond to the
+schematic below:
+
+![GIG schematic](_static/schematic.png)
+
+### Inversions
+
+The easiest example is an inversion. This would be an iedge like
+
+```
+{parent: P, child: C, child_left: 6, child_right: 14, parent_left: 14, parent_right: 6}
+```
+
+There is a subtle gotcha here, because intervals in a GIG, as in _tskit_, are treated as half-closed
+(i.e. do not include the position given by the right coordinate). When we invert an interval, it
+therefore does not include the *left* parent coordinate, but does include the *right* parent coordinate.
+Any transformed position is thus out by one. Or to put it another way, an inversion specified
+by child_left=0, child_right=3, parent_left=3, parent_right=0 transforms the points
+0, 1, 2 to 2, 1, 0: although the *interval* 0, 3 is transformed to 0, 3., the *point* 0 is transformed
+to position 2, not position 3. See
+[here](https://github.com/hyanwong/giglib/issues/41#issuecomment-1858530867)
+for more discussion.
+
+### Duplications
+
+A tandem duplication is represented by two iedges, one for each duplicated region:
+
+```
+{parent: P, child: C, child_left: 10, child_right: 20, parent_left: 10, parent_right: 20}
+{parent: P, child: C, child_left: 20, child_right: 30, parent_left: 10, parent_right: 20}
+```
+
+Or one of the iedges could represent a non-adjacent duplication (e.g. corresponding to a transposable element):
+```
+{parent: P, child: C, child_left: 25, child_right: 35, parent_left: 10, parent_right: 20}
+```
+
+### Deletions
+
+A deletion simply occurs when no material from the parent is transmitted to any of its children (and the coordinate system is shrunk)
+
+```
+# Deletion of parental region from 5-15
+{parent: P, child: C, child_left: 0, child_right: 5, parent_left: 0, parent_right: 5}
+{parent: P, child: C, child_left: 5, child_right: 10, parent_left: 15, parent_right: 20}
+```
+