Knot theory and cluster algebras

TL;DR

This paper links knot theory and cluster algebras by associating cluster structures to knot diagrams, revealing new algebraic invariants and conjecturing a cluster automorphism of order two.

Contribution

It introduces a novel construction connecting knot diagrams to cluster algebras via quivers with potential and modules, and relates the Alexander polynomial to cluster algebra invariants.

Findings

Submodule lattice of modules matches Kauffman states

Alexander polynomial realized as a specialization of F-polynomial

Conjecture of a cluster forming a cluster in the algebra

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.