 Solution

Distinct components of the positively oriented torus link $T_{n,n}$ have linking number one. Since every component also has framing $+1$ relative to the Seifert framing, the surgery linking matrix and hence the intersection form of the surgery trace $W({\hat{L}}_1)$ areIts Smith normal form is $\mathrm{diag}(1,0,\dots ,0)$. The surgery exact sequence, equivalently the kernel and cokernel of $Q_1$, gives

In Kirby calculus, sliding the components over one chosen component diagonalizes the framed link to a split $+1$-framed unknot and $n-1$ zero-framed unknots. Hencewhere $♮$ denotes boundary connected sum.