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Knot theory and cluster algebras

Véronique Bazier-Matte, Ralf Schiffler

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.

Knot theory and cluster algebras

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 , where is the number of crossing points in the knot diagram. We then construct indecomposable modules over the Jacobian algebra of the quiver with potential. For each , 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 -polynomial of , for every . Furthermore, we conjecture that the collection of the 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.

Paper Structure

This paper contains 34 sections, 25 theorems, 77 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Knots and links
  4. The Alexander polynomial
  5. Cluster algebras
  6. Laurent phenomenon and positivity
  7. $F$-polynomials
  8. Quivers with potential
  9. Kauffman states
  10. Poset of Kauffman states
  11. The State polynomial
  12. The Jacobian algebra of a link diagram
  13. Removal of 2-cycles
  14. The link diagram module $T$
  15. A partition of $K_1$
  16. ...and 19 more sections

Key Result

Theorem \oldthetheorem

Let $K$ be a diagram of a prime link. Then, for every segment $i$ of $K$, the Alexander polynomial of $K$ is equal to the specialized $F$-polynomial of the $B$-module $T(i)$. That is

Figures (19)

  • Figure 1: Skein relations for the Alexander polynomial.
  • Figure 2: Kauffman counterclockwise transposition from state $\mathscr{S}$ (left) to state $\mathscr{S}'$ (right).
  • Figure 3: Lattice of Kauffman states of the figure-eight knot.
  • Figure 4: Possible values and specializations at $W = t^{\frac{1}{2}}$ and $B = t^{-\frac{1}{2}}$ for the weight of a segment $j$.
  • Figure 5: The quiver of the figure-eight knot of Example \ref{['ex::kstateslattice']}.
  • ...and 14 more figures

Theorems & Definitions (64)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Laurent Phenomenon
  • Theorem \oldthetheorem: Positivity
  • Example \oldthetheorem
  • Theorem \oldthetheorem: K
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 54 more