This is the first post in a sequence of posts I will release regarding some of the topics I covered in my slides for JMM2024. This is intended to be largely expository and not much is really new mathematics.

Ultimately, we would like to build up towards a connection between quivers and dimer models obtained
through a variant of quiver mutation known as **QP mutation**. This is done by taking a dimer model
$\Gamma$ and orienting its graph dual to obtain the associated **dimer quiver**. It is easy to come
up with dimers where the dimer quiver consists of at least one $2$-cycle. When talking about quiver
mutation specifically, we assume that the quiver has no loops or $2$-cycles. Accordingly, this means
that many dimer quivers are not allowed to be mutated by definition.

This issue can be worked around by introducing a **potential**. Potentials are elements of $kQ/[kQ, kQ]$
where $kQ$ is the path algebra of the quiver and $[kQ, kQ]$ is the subspace of $kQ$ spanned by the
commutators. Then we use a variant of quiver mutation known as **QP mutation** as described by Derksen,
Weymann, and Zelevinsky. QP mutation uses the potential to control which $2$-cycles in the graph are
removed during quiver mutation. In our case, this can be exploited to sensibly mutate dimer quivers,
leading to a new kind of transformation on dimer models known as **$n$-face urban renewal**.

The significance of $n$-face urban renewal is that it corresponds to mutation of quivers with potential in the case of $n = 4$. Furthermore, it also shares various properties with QP mutation (such as being an involution).

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