Boundary slopes of punctured tori in 3-manifolds

Author:
C. McA. Gordon

Journal:
Trans. Amer. Math. Soc. **350** (1998), 1713-1790

MSC (1991):
Primary 57M25; Secondary 57M50

DOI:
https://doi.org/10.1090/S0002-9947-98-01763-2

MathSciNet review:
1390037

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Abstract: Let be an irreducible 3-manifold with a torus boundary component , and suppose that are the boundary slopes on of essential punctured tori in , with their boundaries on . We show that the intersection number of and is at most . Moreover, apart from exactly four explicit manifolds , which contain pairs of essential punctured tori realizing and 6 respectively, we have . It follows immediately that if is atoroidal, while the manifolds obtained by - and -Dehn filling on are toroidal, then , and unless is one of the four examples mentioned above.

Let be the class of 3-manifolds such that is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that , where has a torus component , and . Let be slopes on such that . Then . The exterior of the Whitehead sister link shows that this bound is best possible.

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Additional Information

**C. McA. Gordon**

Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Email:
gordon@math.utexas.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01763-2

Received by editor(s):
March 27, 1995

Received by editor(s) in revised form:
February 26, 1996

Article copyright:
© Copyright 1998
American Mathematical Society