Link splitting deformation of colored Khovanov-Rozansky homology (Q6618014)

The last two decades have witnessed the introduction of powerful homological invariants of knots, links, braids, and tangles, which are connected to classical quantum invariants through a decategorification relationship [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005); \textit{M. Khovanov} and \textit{L. Rozansky}, Fundam. Math. 199, No. 1, 1--91 (2008; Zbl 1145.57009); Geom. Topol. 12, No. 3, 1387--1425 (2008; Zbl 1146.57018); \textit{R. Rouquier}, ``Categorification of the braid groups'', Preprint, \url{arXiv:math/0409593}; Contemp. Math. 406, 137--167 (2006; Zbl 1162.20301)]. These invariants are best understood in the context of differential graded categories, each tangle diagram \(\boldsymbol{D}\) being assigned a chain complex \(\mathcal{Z}(\boldsymbol{D})\) over an additive category, Reidemeister moves between such diagrams being assigned specific chain maps that are invertible up to homotopy, and movies between diagrams that encode certain braid/tangel cobordisms being assigned (generally noninvertible) chain maps that are natural, up to homotopy [\textit{C. Blanchet}, J. Knot Theory Ramifications 19, No. 2, 291--312 (2010; Zbl 1195.57024); \textit{C. L. Caprau}, Algebr. Geom. Topol. 8, No. 2, 729--756 (2008; Zbl 1148.57016); \textit{D. Clark} et al., Geom. Topol. 13, No. 3, 1499--1582 (2009; Zbl 1169.57012); \textit{M. Ehrig} et al., Proc. Lond. Math. Soc. (3) 117, No. 5, 996--1040 (2018; Zbl 1414.57010); \textit{B. Elias} and \textit{D. Krasner}, Homology Homotopy Appl. 12, No. 2, 109--146 (2010; Zbl 1210.57014)]. This paper is concerned with instances of this higher structure.\N\NThis paper shows how the action of the monodromy algebra \(\mathrm{HH}_{\ast }(A_{\mathcal{Z}})\) on \(\mathcal{Z}(\boldsymbol{D} )\) allows of constructing \textit{link splitting deformations} of \(\mathcal{Z}(\boldsymbol{D})\), working explicitly with triply-graded Khovanov-Rozansky homology in the colored case as an extension of the uncolored case in [\textit{E. Gorsky} and \textit{M. Hogancamp}, Geom. Topol. 26, No. 2, 587--678 (2022; Zbl 1508.14005)].\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls background on symmetric functions, including the formalism of symmetric functions in the difference of two alphabets and Haiman determinants [\textit{M. Haiman}, J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)].\N\N\item[\S 3] introduces categorical background, setting up conventions for gradings and homological algebra necessary for the next section.\N\N\item[\S 4] introduces a dg 2-category of curved complexes of singular Soergel bimodules, defining \textit{curved Rickard complexes} as certain special 1-morphisms, establishing the following theorem.\N\NTheorem. The Rickard complex \((\beta_{b})\) associated to a colored braid \(\beta_{b}\) admits a deformation in a curved complex \(\mathcal{Y} C(\beta_{b})\) with \(\Delta e\)-curvature. Such a deformation is unique, up to homotopy equivalence. \N\N\item[\S 5] uses curved complexes of singular Soergel bimodules to construct deformed, colored, triply-graded link homology, establishing that the homology \(\mathcal{Y}H_{\mathrm{KR}}(\boldsymbol{L})\) is an invariant of a framed, oriented, colored link \(\boldsymbol{L}\) up to isomorphism of \(\mathbb{Q}\left[ p_{c,i},v_{c,j}\right] \)-modules.\N\N\item[\S 6] obtains a curved colored skein relation [\textit{M. Hogancamp} et al., ``A skein relation for singular Soergel bimodules'', Preprint, \url{arXiv:2107.08117}].\N\N\item[\S 7] investigates a canonical closed morphism \(\Sigma_{a,b} \in\mathrm{Hom}_{\overline{\mathcal{V}}_{a,b}}^{0}(\mathcal{V} \mathsf{FT}_{a,b},\boldsymbol{1}_{a,b})\) that is the colored analog of the link splitting map from [\textit{M. Hogancamp} et al., ``A skein relation for singular Soergel bimodules'', Preprint, \url{arXiv:2107.08117}].\N\N\item[\S 8] speculates on the extension of the conjecture in [\textit{E. Gorsky} et al., Adv. Math. 378, Article ID 107542, 116 p. (2021; Zbl 1459.57018)] to the colored setting, culminating in Conjecture 8.15.\N\N\item[\S 9] establishes the above conjecture for the positive Hopf link, using the explicit description of the splitting map from \S 7.2.\N\N\item[\S 10] studies the link splitting properties of the deformed colored triply-graded link homology.\N\N\item[The appendix] is concerned with some Hom-space computations. \N\end{itemize}