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The triple linking number of an oriented surface link was defined as ananalogical notion of the linking number of a classical link. We consider acertain $m$-component $T^2$-link ($m \geq 3$) determined from two commutativepure $m$-braids $a$ and $b$. We present the triple linking number of such a$T^2$-link, by using the linking numbers of the closures of $a$ and $b$. Thisgives a lower bound of the triple point number. In some cases, we can determinethe triple point numbers, each of which is a multiple of four.

Triple linking numbers and triple point numbers of certain $T^2$-links