4. The ice numbers $a_{1}, u_{2}, \cdots, a_{9}$ can only take two different values, +1 or -1. Then, the maximum value of the expression $a_{1} a_{5} a_{9}-a_{1} a_{6} a_{8}+a_{2} a_{8} a_{7}-$ $a_{2} a_{4} a_{9}+a_{3} a_{4} a_{8}-a_{3} a_{5} a_{7}$ is $\qquad$.
4. 4 .

Since each term in the sum can only be 11 or -1, the sum is even (the parity of $a-b$ and $a+b$ is the same), so the maximum value of the sum is at most 6. It is easy to see that the sum cannot be 6, because in this case $a_{1} a_{5} a_{9}, a_{2} a_{6} a_{7}, a_{3} a_{6} a_{8}$ should all be +1, making their product equal to +1. However, $a_{1} a_{8} a_{8}, a_{2} a_{4} a_{9}, a_{3} a_{5} a_{8}$ should all be -1, making their product equal to -1.

But in fact, these two products should be equal, so the sum cannot be 6. However, the sum can be 4. For example, when $a_{1}=a_{2}=a_{3}=a_{5}=a_{8}=a_{0}=1, a_{4}=a_{6}=a_{7}=-1$, the sum is equal to 4. In conclusion, the maximum value of the sum is 4.