Arf Invariant

A Link invariant which 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. 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

$$a(K_+)=a(K_-)+l(L_1,L_2),$$ (1)

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

$$\Delta_K(-1)\equiv \cases{ 1\ ({\rm mod\ } 8) & if Arf$(K)=0$\cr 5\ ({\rm mod\ } 8) & if Arf$(K)=1$\cr}$$ (2)

(Jones 1985). For the Jones Polynomial $W_K$ of a Knot $K$,

$${\rm Arf}(K)=W_K(i)$$ (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.

Weisstein, E. W. ``Knots.'' Mathematica notebook Knots.m.