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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?

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up vote 2 down vote accepted

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.)

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You should accept your answer! – Greg Kuperberg Mar 18 '11 at 22:00

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