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The drawn component $L_1$ is an unknot. Its exterior is a solid torus $V$, in which $L_2$ is a pattern of a satellite knot. A meridian of a solid torus of $V$ is ${\ell }_1$, whereas its longitudinal direction is $m_1$. Thus the prescribed gluing sends these directions to $m$ and $\ell$, respectively. This identifies $V$ with a tubular neighborhood of $K$ with its zero Seifert framing.

Removing the pattern before making this identification givesEquivalently, meridionally filling the remaining boundary first removes $L_2$ from the construction and leaves $E(K)\cup V=S^3$; its filling core is the required satellite knot $C(K)$.

In the original link diagram, the two parallel passages through a meridional disk of $V$ run in the same direction; closing them uses a single interchange. The winding number of a satellite pattern is therefore two. Untwisting the surrounding disk shows that $P(U)$ is the unknot (the pattern is the $(2,1)$ cable knot, up to the harmless sign of its single interchange). In particular ${\Delta }_{P(U)}=1$. The Satellite formula for the Alexander polynomial now gives the concise answerHere $\doteq$ allows multiplication by $\pm t^j$. Reversing the winding orientation replaces $t^2$ by $t^{-2}$, giving the same Alexander polynomial of a knot up to such a unit.

The Mayer--Vietoris sequence for the splice identifies the companion meridian with the pattern longitude and kills exactly the relation already killed when passing from $X_{U,P}$ to $X_P$. It therefore gives a canonical isomorphismApply multiplicativity of Turaev torsion to the torus union defining $X_{CP}$, and use part (b) to replace the torsion of $X_{U,P}$. The factors $1-[{\lambda }_U]$ cancel against the peripheral factor from the companion exterior. Usingon the three knot exteriors leavesUnder a one-variable identification, ${\iota }_{\ast }^C$ sends the companion variable to $t^w$, where $w=[U]\in H_1(X_P)\cong \mathbb{Z}$ is the winding number of a satellite pattern. Thus this is also the usual Satellite formula for the Alexander polynomial.

The slope-$4/3$ pattern has winding number $4$, so the Satellite formula for the Alexander polynomial givesUsing the stated torus knot formula in each factor,Up to a Laurent unit this may be simplified to $(1-t)(1-t^{24})/((1-t^8)(1-t^3))$.

Take the untwisted Whitehead double of a trefoil knot. The Whitehead pattern has winding number of a satellite pattern zero and becomes the unknot when its companion is the unknot. The Satellite formula for the Alexander polynomial therefore givesThe Whitehead pattern is geometrically essential in its solid torus, so the satellite knot with nontrivial companion $T_{2,3}$ is nontrivial. Hence $K_4$ is not isotopic to the unknot despite having trivial Alexander polynomial of a knot.

A Whitehead double is the satellite formed with the Whitehead pattern, which has winding number zero and whose image on the unknot is an unknot. The Satellite formula for the Alexander polynomial therefore gives $\Delta (t)\doteq 1$ for every untwisted Whitehead double.