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Injectively immersed tori in branched covers over the figure eight knot

Published online by Cambridge University Press:  20 January 2009

Kerry N. Jones
Kerry N. Jones
Affiliation:
The University of Texas, Austin, TX 78712, U.S.A.
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Abstract

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An algorithm is given for determining presence or absence of injectively (at the fundamental group level) immersed tori (and constructing them, if present) in a branched cover of S3, branched over the figure eight knot, with all branching indices greater than 2. Such tori are important for understanding the topology of 3-manifolds in light of (for example) the Jaco-Shalen–Johannson torus decomposition theorem and the fact that the figure eight knot is universal, i.e., that all 3-manifolds are representable as branched covers of S3, branched over the figure eight knot.

The algorithm is principally geometric in its derivation and graph-theoretic in its operation. It is applied to two examples, one of which has an incompressible torus and the other of which is atoroidal.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 231 - 259
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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