  It was previously shown by the second author that every knot in $S^3$ is
ambient isotopic to one component of a two-component, alternating, hyperbolic
link. In this paper, we define the alternating volume of a knot $K$ to be the
minimum volume of any link $L$ in a natural class of alternating, hyperbolic
links such that $K$ is ambient isotopic to a component of $L$. Our main result
shows that the alternating volume of a knot is coarsely equivalent to the twist
number of a knot.
