# Computing immersed normal surfaces in the figure 8 knot complement

## Author

**Richard Rannard**

## Status

Research Report 97-31

Date: 6 November 1997

## Abstract

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

## Content

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Sydney Mathematics and Statistics