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Given any diagram of a link, we define on the cube of Kauffman's states a"2-complex" whose homology is an invariant of the associated framed links, andsuch that the graded Euler characteristic reproduces the unnormalized Kauffmanbracket. This includes a categorification of brackets skein relation. Then weincorporate the orientation information and get a further complex on the samecube that gives rise to a new invariant homology for oriented links, so thatthe graded Euler characteristic reproduces the unnormalized Jones polynomial inKauffman's version. Finally we clarify the relations between this homology andthe original Khovanov homology of oriented links, extending the well knownrelation between the associated two versions of the Jones polynomial.

Step by step categorification of the Jones polynomial in Kauffman's version