Knot theory and cluster algebras II: The knot cluster

TL;DR

This paper establishes a deep connection between knot theory and cluster algebras by associating a cluster algebra to each knot diagram, linking cluster variables to the Alexander polynomial, and demonstrating symmetry and compatibility properties of related representations.

Contribution

It introduces a novel construction of the knot cluster within cluster algebras and proves the compatibility and symmetry of associated representations, advancing the understanding of knot invariants via algebraic structures.

Findings

Cluster variables match F-polynomials of representations T(i).

Representations T(i) form a compatible cluster.

Symmetry property of T(i) representations proven.

Abstract

To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster x with the property that every cluster variable in x specializes to the Alexander polynomial of K. We call x the knot cluster of A. Furthermore, there exists a cluster automorphism of A of order two that maps the initial cluster to the cluster x. We realize this connection between knot theory and cluster algebras in two ways. In our previous work, we constructed indecomposable representations T(i) of the initial quiver Q of the cluster algebra A. Modulo the removal of 2-cycles, the quiver Q is the incidence quiver of the segments in K, and the representation T(i) of Q is built by taking successive boundaries of K cut open at the i-th segment. The relation to the Alexander polynomial stems from an isomorphism between the submodule lattice of T(i) and the lattice of Kauffman states of K relative to segment i. In the current article, we identify the knot cluster x in A via a sequence of mutations that we construct from a sequence of bigon reductions and generalized Reidemeister III moves on the diagram K. On the level of diagrams, this sequence first reduces K to the Hopf link, then reflects the Hopf link to its mirror image, and finally rebuilds (the mirror image of) K by reversing the reduction. We show that every diagram of a prime link admits such a sequence. We further prove that the cluster variables in x have the same F-polynomials as the representations T(i). This establishes the important fact that our representations T(i) do indeed correspond to cluster variables in A. But it even establishes the much stronger result that these cluster variables are all compatible, in the sense that they form a cluster. We also prove that the representations T(i) have the following symmetry property. For all vertices i,j of Q, the dimension of T(i) at j is equal to the dimension of T(j) at i.