Computing immersed normal surfaces in the figure 8 knot complement


Richard Rannard


Research Report 97-31
Date: 6 November 1997


The theory of (embedded) normal surfaces is a powerful technique in 3-manifold topology. There has been much recent interest in extending the theory to immersed surfaces, in particular to attack the word problem for 3-manifolds. Progress in this area has been hindered by the lack of non-trivial examples. This paper (and the related work [MR]) cover a particular example in depth, using methods which may be generalized. We give detailed information on the existence of immersed surfaces in the figure 8 knot complement using non-trivial computational techniques. After an introduction to the theory, introducing some new concepts, we discuss some strategies for enumerating surfaces of low genus. These have been implemented in software written by the author. The results are tabulated and an unusual example discussed.

Key phrases

3-manifolds. immersed normal surfaces. normal surface theory. algorithms

AMS Subject Classification (1991)

Primary: 57N35
Secondary: 57M50


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Sydney Mathematics and Statistics