{
  "id": "2001.04004",
  "version": "v1",
  "published": "2020-01-12T22:08:06.000Z",
  "updated": "2020-01-12T22:08:06.000Z",
  "title": "Tilting modules arising from knot invariants",
  "authors": [
    "Ralf Schiffler",
    "David Whiting"
  ],
  "comment": "18 pages, 5 Figures",
  "categories": [
    "math.RT",
    "math.CO",
    "math.GT"
  ],
  "abstract": "We construct tilting modules over Jacobian algebras arising from knots. To a two-bridge knot $L[a_1,\\ldots,a_n]$, we associate a quiver $Q$ with potential and its Jacobian algebra $A$. We construct a family of canonical indecomposable $A$-modules $M(i)$, each supported on a different specific subquiver $Q(i)$ of $Q$. Each of the $M(i)$ is expected to parametrize the Jones polynomial of the knot. We study the direct sum $M=\\oplus_iM(i)$ of these indecomposables inside the module category of $A$ as well as in the cluster category. In this paper we consider the special case where the two-bridge knot is given by two parameters $a_1,a_2$. We show that the module $M$ is rigid and $\\tau$-rigid, and we construct a completion of $M$ to a tilting (and $\\tau$-tilting) $A$-module $T$. We show that the endomorphism algebra $\\operatorname{End}_AT$ of $T$ is isomorphic to $A$, and that the mapping $T\\mapsto A[1]$ induces a cluster automorphism of the cluster algebra $\\mathcal{A}(Q)$. This automorphism is of order two. Moreover, we give a mutation sequence that realizes the cluster automorphism. In particular, we show that the quiver $Q$ is mutation equivalent to an acyclic quiver of type $T_{p,q,r}$ (a tree with three branches). This quiver is of finite type if $(a_1,a_2)=(a_1,2), (1,a_2),$ or $(2,3)$, it is tame for $(a_1,a_2)=(2,4)$ or $(3,3)$, and wild otherwise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-12T22:08:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "13F60",
      "57M27"
    ],
    "keywords": [
      "tilting modules arising",
      "knot invariants",
      "jacobian algebra",
      "cluster automorphism",
      "two-bridge knot"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}