  The central discovery of $2d$ conformal theory was holomorphic factorization,
which expressed correlation functions through bilinear combinations of
conformal blocks, which are easily cut and joined without a need to sum over
the entire huge Hilbert space of states. Somewhat similar, when a link diagram
is glued from tangles, the link polynomial is a multilinear combination of {\it
tangle blocks} summed over just a few representations of intermediate states.
This turns to be a powerful approach because the same tangles appear as
constituents of very different knots so that they can be extracted from simpler
cases and used in more complicated ones. So far this method has been
technically developed only in the case of arborescent knots, but, in fact, it
is much more general. We begin a systematic study of tangle blocks by detailed
consideration of some archetypical examples, which actually lead to non-trivial
results, far beyond the reach of other techniques. At the next level, the
tangle calculus is about gluing of tangles, i.e. functorial mappings from ${\rm
Hom}({\rm tangles})$, and its main advantage is an explicit realization of
multiplicative composition structure, which is partly obscured in traditional
knot theory.
