Abstract
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.
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