SPLICE and SWITCH for tangle diagrams in the chemical concrete machine

Knot diagrams can be implemented in the chemical concrete machine.  This result has been already touched in the post  Local FAN-IN eliminates GLOBAL FAN-OUT (II). Here I shall give a short and clear exposition.

1. Oriented knot diagrams.  I shall describe the larger collection of oriented tangle diagrams. A knot is a proper embedding of an oriented circle in 3D space. A tangle is a proper  embedding of a finite collection of  oriented 1D segments (arcs) and circles  in 3D space. A tangle diagram represents a projection of a tangle in general position on a plane (i.e. so that different pieces of arcs or circles do not project on tangent curves in the plane), along with supplementary information at the crossings of the projections (i.e. when the  projections of two pieces of arcs cross, there is more information about which piece passes over).  In this description, a tangle diagram is a 4-valent, locally planar graph, made by two elementary nodes, called “crossings” (usually tangle diagrams are defined as being globally planar graphs made by crossings), and by loops.

chem_cross_0

2. Oriented Reidemeister moves. Two (globally planar) tangle diagrams represent the same 3D tangle up to regular deformations in 3D if and only if we can pass from one tangle diagram to another by a finite sequence of oriented Reidemeister moves (I use here, as previously, the names from   Michael Polyak “Minimal generating sets of Reidemeister moves“, excepting that I use  the letter “R” instead of the letter “\Omega“) .

reidemeister_1

reidemeister_2

reidemeister_3

These are 16 local graph rewrites (local moves) acting on the collection of tangle diagrams.

 3.   My purpose now is to give an implementation of tangle diagrams (not especially globally planar ones, but the more general locally planar ones)  in the chemical concrete machine formalism. For this I have to define the elementary nodes of tangle diagrams as molecules in the chemical concrete machine formalism. This is easy: the loops are the same as in the chemical concrete machine, the crossings are defined as in the following figure:

chem_cross

Each oriented tangle diagram is translated into a molecule from the chemical concrete machine, by using this definition of crossings.  As a consequence, each oriented Reidemeister move becomes a local move in the chemical concrete machine, simply by translating the LHS and RHS tangle diagrams into molecules.

4. I have to prove that these translations of the oriented Reidemeister moves are compatible with the moves from the chemical concrete machine in the following sense: each translation of a Reidemeister move can be obtained as a finite chain of moves from the chemical concrete machine.

It is easier to proceed the other way around, namely to ground  the reasoning in the oriented tangle diagrams universe. I shall translate back some moves from the chemical concrete machine, as seen as acting on oriented tangle diagrams.  See this post at mathoverflow for context, basically it’s almost all we need for the proof, supplemented only by the proof of the SWITCH move, as you shall see.

The graphic beta move in the chemical concrete machine translates as a SPLICE move (it has two variants) over oriented tangle diagrams. The elimination of loops becomes a LOOP move. These moves are described in the next figure.

splice_1

You can find in the mathoverflow post a proof that all oriented Reidemeister moves, with the exception of R2c, R2d, R3a, R3h, are a consequence of a finite number of SPLICE and LOOP moves.

It is left to prove that R2c, R2d, R3a, R3h can be expressed by (translations of) some chains of chemical concrete machine moves. It turns out that we can reduce all these moves to the move R2d by using SPLICE and LOOP, therefore the only thing left is to look at R2d.

chem_r2d

By SPLICE and LOOP, the left hand side of R2d becomes:

chem_r2d_1

So R2d is equivalent with the following SWITCH move:

chem_switch

Finally,  in the post   Local FAN-IN eliminates GLOBAL FAN-OUT (II)   there is a proof that the SWITCH move can be obtained from CO-COMM and FAN-IN moves. The next figure shows this.

chem_switch_2

This concludes the proof that R2d can be expressed by a chain of moves from the chemical concrete machine.

________________________

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