Holes and knots

Hole or knot. The two words are easy to confuse. A system's shape is built from what it leaves free: each free distinction is a point, each pair that takes all four combinations is an edge, each free triple is a filled triangle, and so on. A hole is any gap in that shape. A knot is the smallest kind of hole: a set whose every part is free, but which is not free itself.

a knot: an empty triangle            a hole made by ties: a square

        A                               a ──── b
       / \                              │      │
      B ─ c                             d ──── c

all three pairs are free;            the pairs a–c and b–d are tied, so
A, B and c together are not          those edges are missing; the gap is
                                     made of missing edges

• Pairwise rules make only "flag" shapes. In a 2-CNF system a set is free exactly when all its pairs are free. So two-place rules can make holes, as the square shows, but never an empty triangle. Checked (SpaceTest; of 399 random 2-clause systems on six variables, 29 have holes and none has an empty triangle).
• Every knot is a hole, but not every hole is a knot. To tell whether things are bound, look at the knots, not at the size of the holes. Two unrelated knots side by side leave a three-dimensional hole, because the join of two circles is a 3-sphere, yet nothing binds them.
• Logic allows every shape. Every simplicial complex is the shape of some system. Proved (Lean): every_complex_is_a_nerve in LoopNerve.lean builds one from loops whose contacts permute witness values.

Reading. Logic gives intrinsic holes, paths and separation, but it selects no space. Whatever space there is has to come from a law. Which law gives a shape that looks like a space of low dimension is open (lesson 7.2, question 1).

PYTHONPATH=tools python3 -m knotlab space