Abstract
From a viewpoint of Heegaard theory, we have two types of natural positions of knots in connected closed orientable 3–manifolds: (i) a bridge position with respect to a Heegaard surface, and (ii) a core position of a handlebody bounded by a Heegaard surface. A Heegaard surface of type (ii) corresponds to that of a knot exterior. Hence it has a close connection to Heegaard genus and tunnel number of knots defined below.
Highlights
From a viewpoint of Heegaard theory, we have two types of natural positions of knots in connected closed orientable 3–manifolds: (i) a bridge position with respect to a Heegaard surface, and (ii) a core position of a handlebody bounded by a Heegaard surface
It has a close connection to Heegaard genus and tunnel number of knots defined below
A knot K, that is, a connected closed 1–manifold in M is in a .g; b/–bridge position if K is in a b –bridge position with respect to a Heegaard surface of genus g
Summary
From a viewpoint of Heegaard theory, we have two types of natural positions of knots in connected closed orientable 3–manifolds: (i) a bridge position with respect to a Heegaard surface, and (ii) a core position of a handlebody bounded by a Heegaard surface. Let M be a connected closed orientable 3–manifold and .V1; V2I S / a (genus g ) Heegaard splitting of M , that is, .1/ V1 and V2 are (genus g ) handlebodies, .2/ V1 [ V2 D M and .3/ V1 \ V2 D @V1 D @V2 D S Such a surface S is called a Heegaard surface of M. M admits a .g; b/–bridge position, we obtain a genus g C b Heegaard splitting of M by repeating the converse operation of meridional destabilization and we see hg.K/ Ä g C b. We define meridional destabilizing number of a knot K M as follows: Definition 1.1 Let K be a knot in a connected closed orientable 3–manifold M. Knots in K22 are non-trivial 2–bridge knots, those in K21 are .1; 1/–knots which are not 2–bridge knots, and those in K20 are the other tunnel number one knots
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