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The upsilon invariant at $1$ of $3$–braid knots

Paula Truöl

Algebraic & Geometric Topology 23 (2023) 3763–3804
Abstract

We provide explicit formulas for the integer-valued smooth concordance invariant υ(K) = ΥK(1) for every 3–braid knot K. We determine this invariant, which was defined by Ozsváth, Stipsicz and Szabó (2017), 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) + υ(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.

Keywords
knots, concordance, $3$–braids, upsilon invariant, alternation number, fractional Dehn twist coefficient
Mathematical Subject Classification
Primary: 57K10
Secondary: 20F36, 57K18
References
Publication
Received: 20 September 2021
Revised: 22 March 2022
Accepted: 13 July 2022
Published: 5 November 2023
Authors
Paula Truöl
Department of Mathematics
ETH Zurich
Zurich
Switzerland

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