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The \(\mathfrak{sl}_n\) foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality. (English) Zbl 1360.57025
In [Algebr. Geom. Topol. 4, 1045–1081 (2004; Zbl 1159.57300)], M. Khovanov gave a description of a homology theory in terms of webs and foams categorifying the \(\mathfrak{sl}_3\) link polynomial. [M. Khovanov and L. Rozansky, Fundam. Math. 199, No. 1, 1–91 (2008; Zbl 1145.57009)] introduced a link homology theory categorifying the \(\mathfrak{sl}_n\) link polynomial for \(n\geq 4\).
In the paper under review, the authors give a combinatorial description of colored \(\mathfrak{sl}_n\) link homologies in terms of \(\mathfrak{sl}_n\) webs and foams recovering (colored) Khovanov-Rozansky homology. This improves earlier work of M. Mackaay et al. [Geom. Topol. 13, No. 2, 1075–1128 (2009; Zbl 1202.57017)].
The authors introduce enhanced foam facets fixing sign issues associated with the original matrix factorization formulation and use skew Howe duality to give an algorithm to evaluate close foams combinatorially.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
18G60 Other (co)homology theories (MSC2010)
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