We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine--Tristram signature.

Keywords:

Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism

MSC Classifications:

57M25, 57M27, 18G60 show english descriptions Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Invariants of knots and 3-manifolds
Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22] 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}
57M27 - Invariants of knots and 3-manifolds
18G60 - Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

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