The upsilon invariant at 1 of 3-braid knots
The main purpose of the paper is to determine an explicit formula for the Heegaard Floer-theoretic invariant \(\upsilon\) of all 3-braid knots, which, among its other properties, gives a lower bound for the 4-genus of a knot. The proof strategy relies on the facts that \(\upsilon\) is known for positive torus knots by [\textit{P. S. Ozsváth} et al., Adv. Math. 315, 366--426 (2017; Zbl 1383.57020)] and that the difference in \(\upsilon\) for cobordant knots is bounded by the genus of any cobordism between them. The author first considers positive 3-braid knots and derives upper bounds on the invariant \(\Upsilon\) that subsumes \(\upsilon\), then showing that they are always realised by constructing appropriate cobordisms from the knots in question to connected sums of torus knots. Subsequently, the approach is extended for all 3-braid knots. The proofs of the main result are based on careful calculations in the 3-braid group \(B_3\) that take advantage of both Garside and Murasugi normal forms of elements of \(B_3\). It should be noted that the methods employed in the case of positive 3-braid knots do not in essence depend on the definition of \(\upsilon\) but only on its properties as a homomorphism \(\mathcal{C} \rightarrow \mathbb{Z}\) from the concordance group, as well as the values it takes on torus knots of index \(2\) and \(3\); however, the extension to the non-positive case is somewhat suprising, as it follows from [\textit{P. Feller} et al., ``Squeezed knots'', Preprint, \url{arXiv:2202.12289}] that the Rasmussen \(s\)-invariant and the Heegaard Floer \(\tau\) invariant may not be so determined for the 3-braid knot \(10_{125}\). As an application, various quantities such as the alternation number, the Turaev genus and the minimal number of syllables in a 3-braid word are shown to be determined precisely by \(\upsilon\) and the knot genus for positive 3-braid knots; fractional Dehn twist coefficients are also computed for all 3-braids.
scientific article; zbMATH DE number 7768753
scientific article