Link splitting deformation of colored Khovanov-Rozansky homology
type: Item
instance of: scholarly article
title: Link splitting deformation of colored Khovanov-Rozansky homology
author: Matthew Hogancamp
author: David E. V. Rose
author: Paul Wedrich
published in: Proceedings of the London Mathematical Society. Third Series
publication date: 2024-10-11T00:00:00Z
review text: 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 ca…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 57K18
Mathematics Subject Classification ID: 18N25
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 7925481
MaRDI profile type: Publication
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DOI: 10.1112/PLMS.12620
description: scientific article; zbMATH DE number 7925481