Why does the Jones polynomial never vanish?

Question

Recently, I have been learning the knot-invariants in knot theory. I know the definition and the skein relation of the Jones polynomial. However, I just couldn’t understand why the Jones polynomial never equals zero.

Answer 1

The Jones polynomial $V(L;t)$ satisfies the skein relation $$(t^{1/2} - t^{-1/2}) V(L_0;t) = t^{-1}\;V(L_+;t) - t\; V(L_-;t)$$ where $L_0$, $L_+$, and $L_-$ differ only at a crossing as depicted below.

Substituting $t=1$ into the skein relation yields $V(L_+;1)=V(L_-;1)$. Hence crossing changes do not alter the Jones polynomial of a link evaluated at $t=1$. Therefore if $L$ is any $\ell$-component link, then the Jones polynomial of $L$ evaluated at $t=1$ is the same as the Jones polynomial of the $\ell$-component unlink evaluated at $t=1$.

The $\ell$-component unlink $\bigcirc \sqcup \cdots \sqcup \bigcirc$ has Jones polynomial $V({\bigcirc \sqcup \cdots \sqcup \bigcirc};t) = \left(-t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)^{\ell-1}$. Evaluating at $t=1$ yields $(-2)^{\ell-1}$, which is never zero.

Since the Jones polynomial evaluated at $t=1$ is never zero, it follows that the Jones polynomial itself is never zero.