Surface-alternating knots and links

In this thesis we study several classes of knots and links which have alternating projections onto closed orientable surfaces. We are particularly interested in surface-alternating projections which have a pair of checkerboard surfaces that are topologically essential to the link exterior. We study the structure of generalised alternating links and give a procedure for enumerating their projections. Through the study of their boundary slopes, we can prove some existence results. We introduce a new class of surface-alternating links which we call weakly generalised alternating links. We show that both checkerboard surfaces associated to a weakly generalised alternating diagram are essential in the link exterior. For these links, there are no essential bigons between the two checkerboard surfaces, and we show this is equivalent to the relative $1$-line property. This allows us to give a topological characterisation of weakly generalised alternating link exteriors. We are also able to give a non-diagrammatic characterisation of planar alternating links, which answers a long-standing question of Fox. This means that alternating is a topological property of the link exterior, and not just a diagrammatic property. Finally we use normal surface theory to produce algorithms which can decide if a knot is alternating or weakly generalised alternating, given either a link exterior or a non-alternating planar projection as input.