Arf Invariant
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
![Delta_K(-1)Delta_K(1)=<span style=]() {1 (mod 8) if Arf(K)=0; 5 (mod 8) if Arf(K)=1 " src="https://mathworld.wolfram.com/images/equations/ArfInvariant/NumberedEquation2.gif" style="height:41px; width:258px" /> |
(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.
REFERENCES:
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 223-231, 1994.
Jones, V. "A Polynomial Invariant for Knots via von Neumann Algebras." Bull. Amer. Math. Soc. 12, 103-111, 1985.
Sloane, N. J. A. Sequences A131433 and A131434 in "The On-Line Encyclopedia of Integer Sequences."
