3D crossings in emergent algebras

This post continues the previous one  3D crossings in graphic lambda calculus . Other relevant posts are:

Four smbols

Computing with space: done!

For graphic lambda calculus see this, for knot diagrams and emergent algebras see this, sections 3-6.

In the previous post we saw that we can “construct” crossings   by using both the \lambda abstraction operation and the composition operation from lambda calculus. These operations appear as elementary gates in graphic lambda calculus, along with other two gates, namely the FAN-OUT gate denoted by Y and the \bar{\varepsilon} gate (with \varepsilon an element in a commutative group \Gamma). This last gate models the family of operations of an emergent algebra.

The FAN-OUT gate Y is used in two different contexts. The first one is   as a properly defined FAN-OUT, with behaviour described by the  global fan-out move, needed in the construction which attaches to any term in untyped lambda calculus a graph in the lambda calculus sector of the graphic lambda calculus as explained in “Local and global moves on locally planar trivalent graphs … ” step 3 in section 3. The second one is in relation with the decorated knot formalism of emergent algebras, “Computing with space, …” section 3.1″.

There is an astounding similarity between the \lambda and composition gates from lambda calculus, on one side, and FAN-OUT and \bar{\varepsilon} gates from emergent algebras, in the context of defining crossings. I shall explain this further.

In the decorated knots formalism, crossings of oriented wires are decorated with elements \varepsilon of a commutative group \Gamma. The relation between these crossings and their representations in terms of trivalent graphs is as following:

Comparing with the notations from the previous post, we see that in both cases the \lambda gate corresponds to a FAN-OUT, but, depending on the type of crossing, the composition operation gate corresponds to one of the gates decorated by \varepsilon OR \varepsilon^{-1}.

There is a second remark to be made, namely that the crossings constructed from FAN-OUT and \bar{\varepsilon} gates satisfy Reidemeister I and II moves but not Reidemeister III move. This is not a bad feature of the construction, in fact is the most profound feature, because it leads to the “chora” construction and introduction of composite gates which “in the infinitesimal limit”, satisfy also Reidemeister III, see section 6 from “Computing with space”.

In contradistinction, the crossings constructed from the \lambda abstraction and composition operation do satisfy the three Reidemeister moves.

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