The upsilon invariant at 1 of 3-braid knots
type: Item
instance of: scholarly article
title: The upsilon invariant at 1 of 3-braid knots
arXiv ID: 2108.03674
author: Paula Truöl
published in: Algebraic \& Geometric Topology
publication date: 2023-11-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/2108.03674
review text: The main purpose of the paper is to determine an explicit formula for the Heegaard Floer-theoretic invariant \(\upsilon\) of all 3-braid knots, which, among its other properties, gives a lower bound for the 4-genus of a knot. The proof strategy relies on the facts that \(\upsilon\) is known for positive torus knots by [\textit{P. S. Ozsváth} et al., Adv. Math. 315, 366--426 (2017; Zbl 1383.57020)] and that the difference in \(\upsilon\) for cobordant knots is bounded by the genus of any cobordism between them. The author first considers positive 3-braid knots and derives upper bounds on the invariant \(\Upsilon\) that subsumes \(\upsilon\), then showing that they are always realised by constructing appropriate cobordisms from the knots in question to connected sums of torus knots. Subsequently, the approach is extended for all 3-braid knots. The proofs of the main result are based on careful calculations in the 3-braid group \(B_3\) that take advantage of both Garside and Murasugi normal forms of elements of \(B_3\). It should be noted that the methods employed in the case of positive 3-braid knots do not in essence depend on the definition of \(\upsilon\) but only on its properties as a homomorphism \(\mathcal{C} \rightarrow \mathbb{Z}\) from the concordance group, as well as the values it takes on torus knots of index \(2\) and \(3\); however, the extension to the non-positive case is somewhat suprising, as it follows from [\textit{P. Feller} et al., ``Squeezed knots'', Preprint, \url{arXiv:2202.12289}] that the Rasmussen \(s\)-invariant and the Heegaard Floer \(\tau\) invariant may not be so determined for the 3-braid knot \(10_{125}\). As an application, various quantities such as the alternation number, the Turaev genus and the minimal number of syllables in a 3-braid word are shown to be determined precisely by \(\upsilon\) and the knot genus for positive 3-braid knots; fractional Dehn twist coefficients are also computed for all 3-braids.
reviewed by: Vitalijs Brejevs
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 57K18
zbMATH DE Number: 7768753
zbMATH Keywords: knots
zbMATH Keywords: concordance
zbMATH Keywords: 3-braids
zbMATH Keywords: upsilon invariant
zbMATH Keywords: alternation number
zbMATH Keywords: fractional Dehn twist coefficient
MaRDI profile type: Publication
OpenAlex ID: W3192028754
cites work: AN ESTIMATION OF THE ALTERNATION NUMBER OF A TORUS KNOT
cites work: THE DEALTERNATING NUMBER AND THE ALTERNATION NUMBER OF A CLOSED 3-BRAID
cites work: Almost alternating links
cites work: Positive braids of maximal signature
cites work: Heegaard Floer homology and genus one, one-boundary component open books
cites work: Q3692516
cites work: Q3378434
cites work: Studying links via closed braids. III: Classifying links which are closed 3-braids
cites work: ON QUASI-MORPHISMS FROM KNOT AND BRAID INVARIANTS
cites work: A survey on the Turaev genus of knots
cites work: On the algebraical braid group
cites work: The Jones polynomial and graphs on surfaces
cites work: Turaev genus, knot signature, and the knot homology concordance invariants
cites work: Q4951685
cites work: Optimal cobordisms between torus knots
cites work: Braids with as many full twists as strands realize the braid index
cites work: On cobordisms between knots, braid index, and the upsilon-invariant
cites work: Alternation numbers of torus knots with small braid index
cites work: Knot concordances and alternating knots
cites work: Essential laminations in 3-manifolds
cites work: THE BRAID GROUP AND OTHER GROUPS
cites work: Two bridge knots are alternating knots
cites work: Donaldson's theorem, Heegaard Floer homology, and knots with unknotting number one
cites work: A survey on Heegaard Floer homology and concordance
cites work: Braids, Fibered Knots, and Concordance Questions
cites work: Upper bound for the alternation number of a torus knot
cites work: On alternation numbers of links
cites work: Gauge theory for embedded surfaces. I
cites work: Rasmussen's spectral sequences and the \(\mathfrak{sl}_N\)-concordance invariants
cites work: Computations of the Ozsvath-Szabo knot concordance invariant
cites work: Notes on the knot concordance invariant upsilon
cites work: The Khovanov width of twisted links and closed 3-braids
cites work: Alternating distances of knots and links
cites work: On the Khovanov and knot Floer homologies of quasi-alternating links
cites work: Closed braids which are not prime knots
cites work: Some metrics on classical knots
cites work: On a Certain Numerical Invariant of Link Types
cites work: On closed 3-braids
cites work: Concordance homomorphisms from knot Floer homology
cites work: Unoriented Knot Floer Homology and the Unoriented Four-Ball Genus
cites work: Knot Floer homology and the four-ball genus
cites work: Khovanov homology and the slice genus
cites work: Quasipositivity as an obstruction to sliceness
cites work: Some relations among various numerical invariants for links
cites work: Homology of group systems with applications to knot theory
DOI: 10.2140/AGT.2023.23.3763
description: scientific article; zbMATH DE number 7768753