  The theory of the Kauffman bracket, which describes the Jones polynomial as a
sum over closed circles formed by the planar resolution of vertices in a knot
diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N --
but for a special class of bipartite diagrams made entirely from the
anitparallel lock tangle. Many amusing and important knots and links can be
described in this way, from twist and double braid knots to the celebrated
Kanenobu knots for even parameters -- and for all of them the entire HOMFLY
polynomials possess planar decomposition. This provides an approach to
evaluation of HOMFLY polynomials, which is complementary to the arborescent
calculus, and this opens a new direction to homological techniques, parallel to
Khovanov-Rozansky generalisations of the Kauffman calculus. Moreover, this
planar calculus is also applicable to other symmetric representations beyond
the fundamental one, and to links which are not fully bipartite what is
illustrated by examples of Kanenobu-like links.
