The upsilon invariant at 1 of 3-braid knots

Abstract

We provide explicit formulas for the integer-valued smooth concordance invariant $\upsilon (K)={{\rm Y}}_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsváth, Stipsicz and Szabó, 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.

arXiv comment34 pages, 3 figures. Comments are welcome! V2: Implementation of referee's suggestions. A small error in the uniqueness statement of Prop. 3.2 which is never used or mentioned later in the article was corrected. Accepted for publication in Algebraic & Geometric Topology

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Cited by in corpus (2)

• 3-braid knots with maximal 4-genus
• Exceptional topology on nonorientable manifolds