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 
, 2, ... crossings having Arf invariants 0 and 1 are summarized
in the table below.
 | OEIS | counts
of prime knots with  ,
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, ... |
If 
, 
, and 
are projections which are identical outside the region of
the crossing diagram, and 
and 
are knots
while 
is a 2-component link with a nonintersecting crossing diagram
where the two left and right strands belong to the different links,
then
 |
(1)
|
where 
is the linking number of 
and 
.
The Arf invariant can be determined from the Alexander polynomial or Jones polynomial for a knot.
For 
the Alexander polynomial of 
, the Arf invariant is given by
 |
(2)
|
(Jones 1985). Here, the 
factor takes care of the ambiguity introduced by the fact that the Alexander
polynomial is defined only up to multiples of 
. (Alternately, this indeterminacy is also taken care
of by the Conway definition of the polynomial.)
For the Jones polynomial 
of a knot 
,
 |
(3)
|
(Jones 1985), where i is the imaginary number.
