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A framed link is a link together with a homotopy class of nonzero normal vector field on every component. Relative to the Seifert framing, each component framing is encoded by an integer.

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.