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 n=1, 2, ... crossings having Arf invariants 0 and 1 are summarized in the table below.

Arf(K)OEIScounts of prime knots with n=1, 2, ... crossings
0A1314330, 0, 0, 0, 1, 1, 3, 10, 25, 82, ...
1A1314340, 0, 1, 1, 1, 2, 4, 11, 24, 83, ...

If K_+, K_-, and L are projections which are identical outside the region of the crossing diagram, and K_+ and K_- are knots while l is a 2-component link with a nonintersecting crossing diagram where the two left and right strands belong to the different links, then


where l is the linking number of L_1 and L_2.

The Arf invariant can be determined from the Alexander polynomial or Jones polynomial for a knot. For Delta_K the Alexander polynomial of K, the Arf invariant is given by

 Delta_K(-1)Delta_K(1)={1 (mod 8)   if Arf(K)=0; 5 (mod 8)   if Arf(K)=1

(Jones 1985). Here, the Delta(1) factor takes care of the ambiguity introduced by the fact that the Alexander polynomial is defined only up to multiples of +/-t^i. (Alternately, this indeterminacy is also taken care of by the Conway definition of the polynomial.)

For the Jones polynomial W_K of a knot K,


(Jones 1985), where i is the imaginary number.

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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."

Referenced on Wolfram|Alpha

Arf Invariant

Cite this as:

Weisstein, Eric W. "Arf Invariant." From MathWorld--A Wolfram Web Resource.

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