Upsilon-like concordance invariants from \(\mathfrak{s}\mathfrak{l}_n\) knot cohomology
The authors define new knot concordance invariants from (specific deformations of) Khovanov-Rozansky \(\mathfrak{sl}_n\)-homologies. This invariant, called \(\gimel_n\), is analogous to the Ozsváth-Stipsicz-Szabó \(\Upsilon\) invariant from [\textit{P. S. Ozsváth} et al., Adv. Math. 315, 366--426 (2017; Zbl 1383.57020)]. Furthermore, they provide examples where \(\gimel_n\) can be used to obstruct sliceness and concordance to alternating links, whereas \(\Upsilon\) and other known invariants fail. Finally, the two authors define a new concordance invariant \(S\), which is a direct summand of the equivariant Khovanov-Rozansky complex, called the equivariant Rasmussen invariant. This invariant `dominates' a number of known concordance invariants arising from Khovanov-Rozansky homologies. To sketch the construction of \(\gimel_n\) we start by reviewing the construction of \(\Upsilon\). In knot Floer homology one can `blend' the algebraic and the Alexander filtrations on \(CFK^-(K)\) into a unique filtration which depends on a parameter \(t\in[0,2]\). The least possible filtered degree (in homology) where one can find non-trivial elements gives the value of \(\Upsilon_K(t)\). The construction of the invariant \(\gimel_n\) proceeds in a similar, albeit different, way. Once a potential (i.e. a monic polynomial of degree \(n\)) is fixed one can define a deformation of Khovanov-Rozansky \(\mathfrak{sl}_n\)-homology with respect to the chosen potential. In this case we pick the potential to be \(x^n - x^{n-1}\), then there is a well defined (diagram dependent) chain \(\psi_D\). Similarly to the case of knot Floer homology, one `blends' the quantum and \(x\)-filtration on the deformed complex into a unique, \(t\)-dependent, filtration. The value of \(\gimel_n(K)(t)\) is the least possible filtered degree where the homology class of \([\psi_D]\) does not vanish. Similarly to \(\Upsilon\), the invariant \(\gimel_n\) is a piecewise linear function, and provides a lower bound on the slice genus. However, differently from \(\upsilon\), the function \(\gimel_n\) in \textbf{not} a concordance homomorphism (but a quasi-homomorphism), and provides highly non trivial information for quasi-alternating knots. Finally, towards the end of the paper the authors prove that (i) the stable (i.e. up to acyclic summands) homotopy type of the equivariant Khovanov-Rozansky homology is a link invariant, and (ii) the existence of the equivariant Rasmussen invariant \(S\). Fact (i) is a weaker analogue of the concordance invariance of the knot Floer chain complex up to stable homotopy, due to \textit{J. Hom} [J. Knot Theory Ramifications 26, No. 2, Article ID 1740015, 24 p. (2017; Zbl 1360.57002)]. While fact (ii) is reminescent (and can be seen as a partial extension of) the work of \textit{J. Pardon} concerning the Khovanov-Lee homology of links [Algebr. Geom. Topol. 12, No. 2, 1081--1098 (2012; Zbl 1263.57007)].
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