Geodesic
The upsilon invariant at 1 of 3–braid knots
Truöl, Paula1
1 Department of Mathematics, ETH Zurich, Zurich, Switzerland
Algebraic and Geometric Topology, Tome 23 (2023) no. 8, pp. 3763-3804

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/agt.2023.23.3763
Keywords: knots, concordance, $3$–braids, upsilon invariant, alternation number, fractional Dehn twist coefficient

Truöl, Paula 1

1 Department of Mathematics, ETH Zurich, Zurich, Switzerland 
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Truöl, Paula. The upsilon invariant at 1 of 3–braid knots. Algebraic and Geometric Topology, Tome 23 (2023) no. 8, pp. 3763-3804. doi: 10.2140/agt.2023.23.3763

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