Abstract
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 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, and functorial mappings from Hom(tangles). Its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.
Highlights
The main task of modern quantum field theory is to find and put under control relations between non-perturbative correlation functions, which could provide their Lagrangianindependent description
Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of tangle blocks summed over just a few representations of intermediate states
We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which lead to non-trivial results, far beyond the reach of other techniques
Summary
The main task of modern quantum field theory is to find and put under control relations between non-perturbative correlation functions, which could provide their Lagrangianindependent description. We mostly consider the double-line reducible links and knots which can be separated by cutting, say, a pair of lines (generalization to multiple lines is straightforward) Note that this is much more than just cabling, which is a particular case of the double line going through K1 and K2 without entangling. We provide more examples of tangles of the both types, simple, universal, but R-dependent cut-and-join, which have non-trivial matrix structure, and R-independent cable, which are diagonal matrices and, proportional to the HOMFLY polynomials, which can be calculated by a variety of already developed methods, from cabling and skein relations to arborescent calculus and differential expansions. There would appear additional matrices in the space of multiplicities, and the formulas become too overload
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