| In 1988, Jaco and Rubinstein [7] showed that the normal surfaces of Kneser [9] are an appropriate piecewise linear analogue to index 0 minimal surfaces. Later, Rubinstein introduced the concept of an almost normal surface [15], and showed that these are analogous to index 1 minimal surfaces. The present work is divided into two main chapters. In the first, the concept of an almost normal surface is extended to the case of non-empty boundary. In the second, normal surfaces are generalized further, to include analogues to index 2 minimal surfaces. Applications are presented through, and include answers to algorithmic and finiteness questions for Heegaard splittings, Dehn surgery, and bridge position for knots. |
