Dualizable framed handle link

Assume countable choice. A dualizable framed handle link is an ordered three-component link $(B,G,R)$ in $S^3$ in which $R$ is designated as a dotted component representing a $4$-dimensional $1$-handle, $B$ and $G$ are designated as $0$-framed $2$-handle attaching circles (their framing pushoffs represent their preferred longitudes), the ordered sublink $(B,R)$ is ambient isotopic as an unframed link to $B$ together with a meridian of $B$, the ordered sublink $(G,R)$ has the same property with $G$ in place of $B$, and $\mathrm{lk}(B,G)=0$. Here $\mathrm{lk}(B,G)$ is half the sum of the signs of the mixed crossings in any regular oriented diagram of $B\cup G$; the linking-number lemma guarantees that this is an integer independent of the chosen diagram. The associated handlebody $X(B,G,R)$ has one $0$-handle, the $1$-handle represented by $R$, and the two indicated $2$-handles. Dotted-circle notation means that the unknotted component $R$ specifies a $1$-handle $D^1\times D^3$ attached along its two foot balls $\mathrm{\partial }{D}^1\times D^3$; its belt sphere is $\{0\}\times S^2$. A $0$-framed $2$-handle is $D^2\times D^2$ attached along its attaching region $S^1\times D^2$, with the boundary pushoff representing the preferred longitude.

Piccirillo, §3, definition preceding Theorem 3.1

Hom, §3