COMBINATORICS AND FOUR BRIDGED KNOTS

Abstract

The ℝ P 3-Conjecture states a non-trivial knot in S 3 cannot yield ℝ P 3 by a Dehn surgery. Generically, in the knot-space S3-N(K), the intersection of a projective plane ℝP2 in ℝ P 3, and any 2-sphere S2 in S3 pierced by K, is a 1-complex which can be viewed as a graph in either the projective plane or the 2-sphere. Gordon and Luecke have used similar graphs arising as the intersection of two 2-spheres, to prove that a knot in S3 is determined by its complement. A part of this paper concerns some new combinatorial results on these graphs. They are considered as an unavoidable step towards showing that the ℝ P 3-Conjecture is true. Moreover, we use these results to prove that any non-trivial knot that could yield ℝ P 3 has at least five bridges.