Lesson 2.3 · Chapter 2: The knot test
What a knot hides
Explore all eight states and measure the information hidden from every pair.
From John Cullen’s knotlab guide
Explore the idea
How much can the whole hide?
Click a state to allow or exclude it. The knot test is recomputed after every change. Try to exclude more than half the states while keeping every pair free.
Allowed states
7 / 8
Information in the restriction
0.193 bits
Knot test
Passes
The intuition. A knot is invisible in its parts, so the whole holds information that no part holds. That information can be measured, and it is small: a knot hides at most one bit.
How it is measured. The measure counts possibilities. A knot's pairs are all free, so together they rule nothing out and leave all 8 states possible. The whole allows fewer. If it allows M of the 8 states, it holds log₂(8/M) bits that no pair holds: the atom allows 7 states and holds log₂(8/7) ≈ 0.193 bits, and parity allows 4 and holds log₂(8/4) = 1 bit.
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Two loops and a contact have exactly 34 knots, in five shapes. Checked against all 256 relations on three variables:
Excluded states Relations Bits the whole holds and no pair does 1 (a single three-literal clause: the atom) 8 0.193 2, opposite corners 4 0.415 2, at Hamming distance two 12 0.415 3 8 0.678 4 (parity, even or odd) 2 1.000 A relation on three variables has every pair free exactly when no two of its excluded states differ in one coordinate. So a knot's excluded states form a nonempty independent set of the 3-cube. There are 34 such sets, in five classes under the cube's 48 symmetries.
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A knot hides at most one bit, and only parity hides a full bit. Proved (here). Forgetting one variable of a knot on m variables leaves all 2^(m−1) assignments of the others, and each needs at least one allowed state. So a knot allows at least half of its states. If it allows exactly half, forgetting the last variable is a bijection, so the relation is the graph of a function f: the last variable is fixed by the others. Two allowed states that differ in one coordinate would make that projection two-to-one, so f changes whenever one input bit changes. That makes f parity or its complement.
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The atom is the smallest bond. It excludes the fewest states. It is also what an AND gate adds to its two-place parts: a two-input gate's relation is pairwise exactly when the gate ignores an input, so each of the ten gates that use both inputs needs a three-place clause, which is an atom up to which value of each variable counts as present. Checked (
knotlab worlds).
Reading. A distinction is the material and an atom is what holds material together. Digital hardware can build anything because its gates supply atoms.
PYTHONPATH=tools python3 -m knotlab patterns