The arf invariant is a link invariant that always has the value 0 or 1. A knot has Arf invariant 0 if the knot is "pass equivalent" to the unknot and 1 if it is pass equivalent to the trefoil knot. Arf invariants are implemented in the Wolfram Language as KnotData[knot, ArfInvariant]. The numbers of prime knots on n = 1, 2, ... crossings having Arf invariants 0 and 1 are summarized in the table below. Arf(K) | OEIS | counts of prime knots with n = 1, 2, ... crossings 0 | A131433 | 0, 0, 0, 0, 1, 1, 3, 10, 25, 82, ... 1 | A131434 | 0, 0, 1, 1, 1, 2, 4, 11, 24, 83, ...

KnotData

Cahit Arf