Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

1 Answer 1

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.)

share|cite|improve this answer
You should accept your answer! –  Greg Kuperberg Mar 18 '11 at 22:00

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.