MATRIX FACTORIZATIONS AND DOUBLE LINE IN sln QUANTUM LINK INVARIANT

L. Kauffman introduced a graphical link invariant which is the normalized Jones polynomial [4][5]. It is well-known that the polynomial is derived from the fundamental representation of the quantum group Uq(sl2). Further, G. Kuperberg constructed a graphical link invariant associated with the fundamental representation of the quantum group Uq(sl3) [10]. H. Murakami, T. Ohtsuki and S. Yamada introduced a graphical regular link invariant for the fundamental representation of the quantum group Uq(sln) [11] . In general, we can also obtain a graphical link invariant for a given quantum group Uq(g) and the fundamental representation. These invariants are collectively called g quantum link invariants. M. Khovanov constructed a categorification of sl2 quantum link invariant [6]. A categorification generally means the replacement of a set with a category by corresponding an element to an object. The morphism of the category is properly chosen to carry theory well-done. For a categorification, there is an inverse operation called a decategorification which is the replacement of a category with a set. The decategorification of equivalent objects in the category is a same element in the set. The Khovanov’s theory is a beautiful example of the categorification; this is the replacement of Jones polynomial J(L), which is a map from the set of links to “Z[q, q]” , by a map CK from the set of links to “the homotopy category of the bounded complex of graded Z-modules”. The bounded complex CK(L) and the Z⊕Zgraded homology groups KH(L) associated with CK(L) also become link invariants under the Reidemeister moves. The decategorification is a Z-graded Euler characteristic χKH with the normalized Jones polynomial J(L) ;