Jones (1985) defines a simplified trace invariant for knots by $W\_K(t)=\frac{1-V\_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W\_K(i)$. Does this work for oriented links as well? If not, is there a variant that does?



Simplified Jones trace invariant for links
Oops, my bad. Since the Arf invariant is not defined for all links, what I was looking for doesn't exist. If the Jones polynomial of an oriented link $L$ in $S^3$ equals 0 when evaluated at $t=\imath$, then the Arf invariant of $L$ is not defined. (See W.B. Likorish, \textit{An Introduction to Knot Theory}, p. 106.)


