What networks of loops remember

From rings to networks. Networks extend this. Put a witness at each vertex of a graph, and let each edge carry permutation contact: a permutation of the witness values takes the witness at one end to the witness at the other. A ring is the case of a single cycle. Composing the permutations around a closed walk gives the walk's holonomy, and a value that a permutation leaves unchanged is one of its fixed points. A loop can also be given a selector, so that it either passes the witness unchanged or applies its holonomy, and a set of loops is switched on together when all their selectors can be on at once. The nerve of a family of sets is the shape with one point for each set and a face for each group of sets that share an element.

Proved (Lean): LoopNerve.lean.

• Free contact records each loop separately. When a contact's every holonomy that fixes one value fixes all of them, as for parity or addition mod q, no two loops ever tie (free_contact_binds_nothing). Each independent loop then carries its own class, a first cohomology class.
• Contact with fixed points binds loops, into atoms. The loops a network can switch on together form the nerve of the holonomies' fixed-point sets (shape_is_nerve). A minimal set of loops that cannot all be on is a knot among loops that excludes exactly one pattern: an atom (minimal_nonface_is_knot).
• Binding costs witness values. A knot among k loops needs k + 1 witness values (knot_needs_more_values). Boolean witnesses cannot even tie two loops (boolean_contact_never_ties).

The first cohomology class of the first point amounts to one value of the contact's group for each independent loop of the network.

Classical background. Composing tables is the transfer-matrix method. A parity ring with an odd number of swaps is Tseitin's parity contradiction on a cycle. Local consistency methods fail exactly on constraints that can simulate equations over a group (Barto and Kozik 2009), the same dividing line. The mathematics here is elementary. The reading is ours.

Reading. The two extreme knots play opposite roles. Parity, which hides the most, carries a single loop at full strength. The atom, which hides the least, binds loops together through fixed points. A law can keep a lasting record of its own loops only if its contacts act as group elements.

PYTHONPATH=tools python3 -m knotlab holonomy