On the Upsilon invariant of fibered knots and right-veering open books (Q2079515)

Given a fibered knot \(K\) in a closed \(3\)-manifold \(Y\), together with the fibration over the circle \(S^1\) of its complement, the construction due to [\textit{W. P. Thurston} and \textit{H. E. Winkelnkemper}, Proc. Am. Math. Soc. 52, 345--347 (1975; Zbl 0312.53028)] gives a contact structure \(\xi_K\) on \(Y\), which is supported by the open book given by \(K\), as defined in [\textit{E. Giroux}, in: Proceedings of the international congress of mathematicians, ICM 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 405--414 (2002; Zbl 1015.53049)]. In this context, one can sometimes effectively use knot Floer techniques on \(K\) to detect properties of the contact structure \(\xi_K\). For instance, it is proven in [\textit{M. Hedden}, J. Knot Theory Ramifications 19, No. 5, 617--629 (2010; Zbl 1195.57029)] that, for fibered knots \(K\) in the \(3\)-sphere \(S^3\), \(\xi_K\) is tight if and only if the \(\tau\)-invariant of \(K\), defined in [\textit{P. Ozsváth} and \textit{Z. Szabó}, Geom. Topol. 7, 615--639 (2003; Zbl 1037.57027)], equals the genus of the fibered surface bounded by \(K\). More generally, according to the aforementioned work by Giroux, any contact structure \(\xi\) on a \(3\)-manifold is supported by an open book, which is moreover unique up to positive stabilizations by work of Giroux-Mohsen. It is then a natural (and hard) problem to understand how the properties of \(\xi\) are determined by and determine those of the supporting open books. For instance, [\textit{K. Honda} et al., Invent. Math. 169, No. 2, 427--449 (2007; Zbl 1167.57008)] proved that a contact structure \(\xi\) is tight if and only if the monodromy \(\varphi\) of any open book supporting \(\xi\) is right-veering. Here, ``right-veering'' roughly means that \(\varphi\) sends every properly embedded arc in the page to the right of itself locally near the endpoints, after an isotopy removing unnecessary intersections. Going in this direction, the paper under review studies connections between some invariants of fibered knots \(K\) coming from knot Floer techniques and properties of the monodromies \(\varphi_K\) of the open book on \(S^3\) determined by \(K\). More precisely, the main result is that, for a null-homologous fibered knot \(K\) in a rational homology sphere \(Y\), if \(\Upsilon_{K,\mathfrak{s}}'(t)=-g\) for some \(t\in[0,1)\), where \(g\) is the genus of the fibered surface, then the open book monodromy \(\phi_K\) is right-veering. Here, \(\Upsilon_{K,\mathfrak{s}}(t)\) denotes the \(1\)-parameter family of concordance invariants of [\textit{A. Alfieri} et al., Stud. Sci. Math. Hung. 58, No. 4, 457--488 (2021; Zbl 1499.57010)] associated to null-homologous knots in rational homology spheres, which is a generalization of the invariant \(\Upsilon_K(t)\) defined for knots \(K\) in \(S^3\) by [\textit{P. S. Ozsváth} et al., Adv. Math. 315, 366--426 (2017; Zbl 1383.57020)]. This result is not enough to conclude tightness of \(\xi_K\) due to the fact that it does not say that \textit{every} supporting open book has right-veering monodromy, but it is important to point out that, if one is only interested in understanding the monodromy of the open book determined by \(K\), it is a stronger result than the one due to Hedden mentioned above. Indeed, the authors give an infinite family of examples of knots in \(S^3\) satisfying the assumption of their Theorem but for which the supported contact structure is overtwisted (hence in particular the \(\tau\) invariant doesn't satisfy the hypothesis of Hedden's Theorem). The second main result in the paper under review concerns concordance of fibered knots. More precisely, the authors prove that any fibered knot \(K\) of genus \(g\) in \(S^3\) such that \(\Upsilon_K'(t)=-g\) for some \(t\in[0,1)\) cannot be concordant to another fibered knot of the same genus and whose monodromy is \textit{not} right-veering. The interest in this result arises from the fact that, without condition on the \(\Upsilon_K\)-invariant, the statement is known to be false. As the authors point out, it is then a question deserving more exploration to better understand how to obstruct concordance between fibered knots efficiently in terms of the associated open book monodromies.