  A correspondence, by way of Heegaard splittings, between closed oriented
3-manifolds and pairs of surjections from a surface group to a free group has
been studied by Stallings, Jaco, and Hempel. This correspondence, by way of
trisections, was recently extended by Abrams, Gay, and Kirby to the case of
smooth, closed, connected, oriented 4-manifolds. We unify these perspectives
and generalize this correspondence to the case of links in closed oriented
3-manifolds and links of knotted surfaces in smooth, closed, connected,
oriented 4-manifolds. The algebraic manifestations of these four subfields of
low-dimensional topology (3-manifolds, 4-manifolds, knot theory, and knotted
surface theory) are all strikingly similar, and this correspondence perhaps
elucidates some unique character of low-dimensional topology.
