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Deformations of colored \(\mathfrak{sl}_N\) link homologies via foams. (English) Zbl 1420.57044
Summary: We prove a conjectured decomposition of deformed \(\mathfrak{sl}_{N}\) link homology, as well as an extension to the case of colored links, generalizing results of E. S. Lee [Adv. Math. 197, 554–586 (2005; Zbl 1080.57015)], B. Gornik [“Note on Khovanov link homology”, Preprint, arXiv:math/0402266], and H. Wu [J. Knot Theory Ramifications 21, No. 2, Article ID 1250012, 104 p. (2012; Zbl 1245.57018)]. To this end, we use foam technology to give a completely combinatorial construction of Wu’s deformed colored \(\mathfrak{sl}_{N}\) link homologies. By studying the underlying deformed higher representation-theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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