Lesson 1.5 · Chapter 1: Bits and rules
Reading A, B → c
The atom forbids exactly one state: A and B without c. Every other state is allowed.
From John Cullen’s knotlab guide
Explore the idea
Every reading forbids the same state
The atom can be read from any of its letters. Pick a reading. An “if … then” reading asks something only of the states where its “if” part holds, which are outlined, and forbids the one among them where the “then” part fails.
States in the order A, B, c
The “if” part holds in 2 states, and the “then” part fails in one of them: 110. 7 of 8 states are allowed.
Ways of writing it. The smallest rule that binds two loops at a contact is called the atom. It is written in several ways, and they all say the same thing:
| Written | Read aloud |
|---|---|
| A ∧ B → c, often shortened to A, B → c | if A and B, then c |
| ¬A ∨ ¬B ∨ c | not A, or not B, or c |
~A ~B c | the same clause, typed for knotlab |
The one thing to remember is that the atom forbids exactly one state: A = 1, B = 1 and c = 0, which is A and B without c. In words, A and B together demand c. Every other state is allowed:
A B c
0 0 0 allowed
0 0 1 allowed
0 1 0 allowed
0 1 1 allowed
1 0 0 allowed
1 0 1 allowed
1 1 0 excluded: A and B without c
1 1 1 allowed
Each piece of the notation, in turn:
- A and B are the open distinctions of the two loops (lesson 1.4), and c is the contact. The capital letters mark the variables that come from loops and the small letter marks the contact, nothing more.
- The comma in A, B → c means "and", as in A ∧ B.
- The arrow is the "if …, then" of lesson 1.2. It rules out only the case in which A and B hold and c does not. It says nothing about c when A or B is 0, so c = 1 with A = B = 0 is allowed. It does not mean that A and B come first, or that they cause c or compute it.
- The direction is only a way of reading it. The clause ¬A ∨ ¬B ∨ c can be read from any of its letters: "if A and B, then c", "if A and not c, then not B", or "if B and not c, then not A". All three rule out the same state, 110. The knot test of lesson 2.1 treats the three variables alike.
Why it binds. Check the three pairs, with "free" as in lesson 1.2.
- A and B take all four combinations of values, since c = 1 completes each of them to an allowed state.
- A and c take all four, since B = 0 completes each of them.
- B and c take all four, since A = 0 completes each of them.
So every pair is free, but the three together are not, since 110 never occurs. The rule lives in the whole and in no part, which is what this guide means by a binding. Lesson 2.1 turns that into the knot test.
Two rules look similar but are not the atom:
- The AND gate c = A ∧ B makes c exactly "A and B". It rules out four states, 001, 011, 101 and 110, and its pair A, c is already constrained, since A = 0 with c = 1 never occurs. The atom asks only that A and B together bring c.
- The common hub (¬A ∨ c) ∧ (¬B ∨ c), "A brings c, and B brings c", rules out 010, 100 and 110. Each loop forces c on its own, so the pair A, c is constrained: A = 1 with c = 0 never occurs. Lesson 2.1 shows why the hub fails the knot test.
Lesson 4.2 reads 1 as "exists". Then A, B → c is a requirement: when A and B exist, c must exist too, and nothing is required otherwise.