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Computing immersed normal surfaces in the figure-eight knot complement. (English) Zbl 0927.57020

Computational methods are applied to determine data on the existence of immersed normal surfaces in the figure-eight knot complement. The problem of enumerating surfaces is reduced to enumerating the elements of a permutation group which have certain properties. This problem is transformed into the question of searching a certain graph, for which various strategies are compared. A table enumerating the number of regular surfaces, connected regular surfaces, and orientable regular surfaces in each class is presented. In one class, there is only one regular surface out of nearly 2 billion possibilities. This surface is shown to have the property that each edge disk is of the same type.

MSC:

57N35 Embeddings and immersions in topological manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds

Software:

Magma
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References:

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