  We establish a connection between knot theory and cluster algebras via
representation theory. To every knot diagram (or link diagram), we associate a
cluster algebra by constructing a quiver with potential. The rank of the
cluster algebra is $2n$, where $n$ is the number of crossing points in the knot
diagram. We then construct $2n$ indecomposable modules $T(i)$ over the Jacobian
algebra of the quiver with potential. For each $T(i)$, we show that the
submodule lattice is isomorphic to the corresponding lattice of Kauffman
states. We then give a realization of the Alexander polynomial of the knot as a
specialization of the $F$-polynomial of $T(i)$, for every $i$. Furthermore, we
conjecture that the collection of the $T(i)$ forms a cluster in the cluster
algebra whose quiver is isomorphic to the opposite of the initial quiver, and
that the resulting cluster automorphism is of order two.
