11. Let the line $y=k x+b$ intersect the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and $|A B|=|B C|=2$. Then the value of $k$ is
11. 1.

Given that the curve is symmetric about the point $(0,0)$, and
$$
|A B|=|B C|=2 \text {, }
$$

we know that the line $y=k x+b$ must pass through the origin.
Thus, $b=0$.
Let $A(x, y)$. Then
$$
\begin{array}{l}
y=k x, y=x^{3}-x, \sqrt{x^{2}+y^{2}}=2 \\
\Rightarrow x=\sqrt{k+1}, y=k \sqrt{k+1} . \\
\text { Substituting into } \sqrt{x^{2}+y^{2}}=2 \text { gives } \\
(k+1)+k^{2}(k+1)=4 \\
\Rightarrow(k-1)\left(k^{2}+2 k+3\right)=0 \\
\Rightarrow k=1 .
\end{array}
$$