Abstract
Using Bar-Natan?s link homology we define a homology theory for framed links whose components are labelled by irreducible representations of the group Uq(sl2). We then compute this explicitly.
Highlights
Khovanov [2005] defined a link homology theory categorifying the coloured Jones polynomial. He constructed a cochain complex associated to an oriented framed link whose components are labelled by irreducible representations of Uq with the property that the graded Euler characteristic of the homology of this complex is the coloured Jones polynomial
J (K n−2i ), i =0 where K j is the j-cable of the knot K, as the Euler characteristic of a complex involving the link homology of the cablings K n−2i, for i = 0, . . . , n/2
Cabling a knot or link immediately introduces an unmanageable number of crossings from a computational point of view
Summary
Khovanov [2005] defined a link homology theory categorifying the coloured Jones polynomial He constructed a cochain complex associated to an oriented framed link whose components are labelled by irreducible representations of Uq (sl2) with the property that the graded Euler characteristic of the homology of this complex is the coloured Jones polynomial. We use the singly graded (filtered) theory, which is defined over ކ2, obtained by setting the variable H in that paper to be 1 This theory is somewhat like Lee’s deformation of Khovanov homology and was computed explicitly in [Turner 2004] following.
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