Tangle blocks in the theory of link invariants

TL;DR

This paper introduces the concept of tangle blocks in link invariants, demonstrating their power in decomposing and analyzing complex knots through a systematic and functorial approach inspired by conformal field theory.

Contribution

It develops a systematic study of tangle blocks beyond arborescent knots, providing explicit methods for their composition and revealing new insights into link invariants.

Findings

Tangle blocks can be used to decompose complex knots into simpler components.

Explicit functorial mappings for gluing tangles are constructed.

The approach extends beyond arborescent knots to more general cases.

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 {\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.