  Checkerboard surfaces in alternating link complements are used frequently to
determine information about the link. However, when many crossings are added to
a single twist region of a link diagram, the geometry of the link complement
stabilizes (approaches a geometric limit), but a corresponding checkerboard
surface increases in complexity with crossing number. In this paper, we
generalize checkerboard surfaces to certain immersed surfaces, called twisted
checkerboard surfaces, whose geometry better reflects that of the alternating
link in many cases. We describe the surfaces, show that they are essential in
the complement of an alternating link, and discuss their properties, including
an analysis of homotopy classes of arcs on the surfaces in the link complement.
