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

(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."