- The basic observation is that 2-tangles are 2-tangles in both "s- and t-channels". Product and sum can be defined for all tangles but only in the case of 2-tangles the sum, which in this case reduces to product in t-channel obtained by putting tangles in series, gives 2-tangle. The sum of M- and N-tangles is M+N-2-tangle and combines various N-tangles to a monoidal structure. Tensor product like operation giving M+N tangle looks to me physically more natural than the sum.
- The reason why general 2-tangles are non-commutative although 2-braids obviously commute is that 2-tangles can be regarded as sequences of N-tangles with 2-tangles appearing only as the initial and final state: N is actually even for intermediate states. Since N>2-braid groups are non-commutative, non-commutativity results. It would be interesting to know whether braid group representations have been used to construct representations of N-tangles.
1. Tangles as number theoretic braids?
The strands of number theoretical N-braids correspond to roots of N:th order polynomial and if one allows time evolutions of partonic 2-surface leading to the disappearance or appearance of real roots N-tangles become possible. This however means continuous evolution of roots so that the coefficients of polynomials defining the partonic 2-surface can be rational only in initial and final state but not in all intermediate "virtual" states.
2. Tangles as tangled partonic 2-surfaces?
Tangles could appear in TGD also in second manner.
- Partonic 2-surfaces are sub-manifolds of a 3-D section of space-time surface. If partonic 2-surfaces have genus g>0 the handles can become knotted and linked and one obtains besides ordinary knots and links more general knots and links in which circle is replaced by figure eight and its generalizations obtained by adding more circles (eyeglasses for N-eyed creatures;-)).
- Since these 2-surfaces are space-like, the resulting structures are indeed tangles rather than only braids. Tangles made of strands with fixed ends would result by allowing spherical partons elongate to long strands with fixed ends. DNA tangles would the basic example, and are discussed also in the article. DNA sequences to which I have speculatively assigned invisible (dark) braid structures might be seen in this context as space-like "written language representations" of genetic programs represented as number theoretic braids.