{
  "id": "2108.03674",
  "version": "v1",
  "published": "2021-08-08T16:02:34.000Z",
  "updated": "2021-08-08T16:02:34.000Z",
  "title": "The upsilon invariant at 1 of 3-braid knots",
  "authors": [
    "Paula Truöl"
  ],
  "comment": "33 pages, 3 figures. Comments are welcome!",
  "categories": [
    "math.GT"
  ],
  "abstract": "We provide explicit formulas for the integer-valued smooth concordance invariant $\\upsilon(K) = \\Upsilon_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsv\\'ath, Stipsicz and Szab\\'o, by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots $K$ several alternating distances all equal the sum $g(K) + \\upsilon(K)$, where $g(K)$ denotes the 3-genus of $K$. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-08T16:02:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K18",
      "20F36"
    ],
    "keywords": [
      "upsilon invariant",
      "alternation number",
      "integer-valued smooth concordance invariant",
      "dealternating number",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}