The Unknotting Problem and Normal Surface Q-Theory

Details The Unknotting Problem and Normal Surface Q-Theory The Unknotting Problem and Normal Surface Q-Theory

2010-09-08

arxiv.org/abs/1009.1500v1

Mathematics

Geometric Topology

13 pages, 4 figures

Scientific paper

Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose $M$ is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if $M$ contains an essential surface, then the projective solution space has an essential surface at a vertex. One interesting situation not covered by this theorem is when $M$ is boundary reducible, e.g. $M$ is an unknot complement. We prove that in this case $M$ has an essential disc at a vertex of the Q-projective solution space.

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