Heegaard structures of manifolds in the Dehn filling space

AbstractWe prove that after Dehn filling an incompressible torus in the boundary of an a-cylindrical 3-manifold the Heegaard genus degenerates by at most one for all but finitely many fillings. We do so by proving that for all but finitely many fillings the core of the attached solid torus can be isotoped into the minimal Heegaard surface of the filled manifold, we say that these manifolds are good. For these fillings, after stabilizing the Heegaard surface once, it becomes a Heegaard surface of the original manifold. We show that any two Heegaard surfaces in different fillings, into which the core is not isotopic, can be isotoped to intersect essentially. Using this, a bound on the distance between fillings containing such surfaces is given in terms of the genera of the Heegaard surfaces

Genus Reducing Knots in 3-Manifolds

A genus reducing knot is a knot that has infinitely many surgeries after which the Heegaard genus of the manifold reduces. We study certain aspects of this question, in particular solving it for totally orientable Seifert Fibered Spaces, where we find examples of manifolds of arbitrarily high genus containing no such knot

Thin position for a connected sum of small knots

We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-14.abs.htm

Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.Comment: Final version. To appear in European Journal of Combinatoric