The upsilon invariant at 1 of 3-braid knots

TL;DR

This paper derives explicit formulas for the upsilon invariant at 1 for all 3-braid knots, linking it to various knot invariants and providing bounds for alternating distances.

Contribution

It introduces explicit formulas for the upsilon invariant at 1 for 3-braid knots and relates it to other knot invariants, enhancing understanding of their concordance properties.

Findings

Computed upsilon invariant for all 3-braid knots.

Established equalities between alternating distances and knot invariants.

Provided bounds on alternation and dealternating numbers for 3-braid knots.

Abstract

We provide explicit formulas for the integer-valued smooth concordance invariant $\upsilon(K) = \Upsilon_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsv\'ath, Stipsicz and Szab\'o, 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.