{
  "id": "2210.16452",
  "version": "v1",
  "published": "2022-10-29T00:42:16.000Z",
  "updated": "2022-10-29T00:42:16.000Z",
  "title": "Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus",
  "authors": [
    "David Boozer"
  ],
  "comment": "60 pages, 26 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,\\delta)$ in the $A_\\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,\\delta)$ is an isotopy invariant of the tangle diagram. We use $(X,\\delta)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \\times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \\times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-29T00:42:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57R58",
      "53D40"
    ],
    "keywords": [
      "fukaya category",
      "traceless character variety",
      "twice-punctured torus",
      "cochain complex reproduces reduced khovanov",
      "complex reproduces reduced khovanov homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}