The upsilon invariant at 1 of 3–braid knots

Truöl, Paula

doi:10.2140/agt.2023.23.3763

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

Paula Truöl

Algebraic & Geometric Topology 23 (2023) 3763–3804

DOI: 10.2140/agt.2023.23.3763

Abstract

We provide explicit formulas for the integer-valued smooth concordance invariant $υ (K) = Υ_{K} (1)$ 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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Volume 23, issue 8 (2023)