2.6.1 Generalized tangles

Generalized n-tangles with 2n emerging arcs instead of 4 (n=2) are mentioned in Conway's paper (1970) without a further elaboration. The same holds for Murasugi (1996, page 172): after defining n-tangles (called there (n,n)-tangles) and giving a sample of a (3,3)-tangle, the author continued to work exclusively with 2-tangles (or (2,2)-tangles). Even the examples of particular n-tangles and their use is hard to find. However, we are very familiar with an elementary 3-tangle: it is the standard illustration of the third Reidemeister move. A. Caudron (1982) used hyperbolic n-tangles for the construction of basic polyhedra, and H. Moriuchi (2004) enumerated theta-curves (i.e., non-algebraic 2-tangles) with up to 7 crossings. Non-algebraic 2-tangles with n 11 crossings are considered in preceding section.

Generalized tangles will be used in order to extend Conway notation to non-algebraic tangles and basic polyhedra, describe non-algebraic tangle types and use a type-algebra for computing the number of components of non-algebraic KLs.