FINDING PLANAR SURFACES IN KNOT- AND LINK-MANIFOLDS

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. Two similar results are obtained with added boundary conditions. Namely, given a link-manifold M, a component B of ∂M, and a slope γ on B, there is an algorithm to decide if there is an embedded punctured-disk in M with boundary γ and punctures in ∂M\B; and given a link-manifold M, a component B of ∂M, and a meridian slope μ on B, there is an algorithm to decide if there is an embedded punctured-disk with boundary a longitude on B and punctures in ∂M\B. In both cases, if there is one, the algorithm will construct one. The proofs introduce a number of new methods and differ from the classical proofs, using normal surfaces, as solutions may not be found among the fundamental solutions.

[1]  J. Rubinstein,et al.  0-Efficient Triangulations of 3-Manifolds , 2002, math/0207158.

[2]  E. Sedgwick,et al.  Decision problems in the space of Dehn fillings , 1998, math/9811031.

[3]  William Kazez,et al.  Polyhedral minimal surfaces, Heegaard splittings, and decision problems for 3-dimensional manifolds , 1996 .

[4]  M. Scharlemann,et al.  Least weight injective surfaces are fundamental , 1996 .

[5]  J. L. Tollefson.  Isotopy classes of incompressible surfaces in irreducible 3-manifolds , 1995 .

[6]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .

[7]  Abigail Thompson,et al.  Thin Position and the Recognition Problem for $\bold{S^3}$ , 1994 .

[8]  Abigail Thompson,et al.  THIN POSITION AND THE RECOGNITION PROBLEM FOR S3 , 1994 .

[9]  Geoffrey Hemion.  The Classification of Knots and 3-Dimensional Spaces , 1992 .

[10]  J. Hyam Rubinstein,et al.  PL equivariant surgery and invariant decompositions of 3-manifolds , 1989 .

[11]  Ulrich Oertel,et al.  An algorithm to decide if a 3-manifold is a Haken manifold , 1984 .

[12]  A. Hatcher,et al.  Boundary Curves of Incompressible Surfaces , 1982 .

[13]  Friedhelm Waldhausen,et al.  The word problem in fundamental groups of sufficiently large irreducible 3-manifolds , 1968 .

[14]  Thomas L. Saaty,et al.  Lectures on modern mathematics , 1963 .

[15]  Horst Schubert,et al.  Bestimmung der Primfaktorzerlegung von Verkettungen , 1961 .

[16]  W. Haken.  Theorie der Normalflächen , 1961 .