Dehn fillings of 3-manifolds and non-persistent tori

✍ Scribed by Luis Gerardo Valdez Sánchez

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 275 KB
• Volume: 98
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00038-3

✦ Synopsis

We prove that if M is a connected, compact, orientable, irreducible 3-manifold with incompressible torus boundary and r is a planar boundary slope in ∂M, then either M contains an essential non-persistent torus with boundary slope r, or a closed essential torus which compresses in the Dehn filling M(r), or M(r) is 'small' (meaning M(r) is a manifold of the form L#L , where each factor L, L is either S 3 , S 1 × S 2 , or a lens space). We also give an example of an infinite family of hyperbolic manifolds with torus boundary, each of which contains two essential non-persistent punctured tori of distinct boundary slopes associated to two reducible Dehn fillings. The main result may also be applied to give a condition under which a knot in S 3 with a reducible surgery must be cabled. We further study the reducible surgeries on knots in S 3 , and show that each prime factor in such a surgery is, with at most one exception, a lens space.