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The regular projective solution space of the figure-eight knot complement. (English) Zbl 0984.57006

The authors analyze clearly, and in some detail, the possibilities for normal immersions of a surface in the complement of the figure-eight knot. They utilize the canonical ideal triangulation of W. P. Thurston [Three-dimensional geometry and topology, Vol. 1, Princeton Math. Ser. 35 (1997; Zbl 0873.57001)], and study carefully the classes of solutions of the linear equations which arise from the conbinatorics of the intersection of a surface, normally immersed in the space, with the 2 tetrahedra comprising the triangulation.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0873.57001

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References:

[1] Aitchison I. R., J. Knot Theory Ramifications 7 (8) pp 1005– (1998) · Zbl 0924.57018 · doi:10.1142/S0218216598000541
[2] DOI: 10.1006/jsco.1996.0125 · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125
[3] Hemion G., The classification of knots and 3-dimensional spaces (1992) · Zbl 0771.57001
[4] Jaco, W. 1987. [Jaco 1987], unpublished personal notes
[5] DOI: 10.1016/0040-9383(84)90039-9 · Zbl 0545.57003 · doi:10.1016/0040-9383(84)90039-9
[6] Jaco W., Illinois J. Math. 39 (3) pp 358– (1995)
[7] Letscher D., Ph.D. thesis, in: Immersed normal surfaces and decision problems for 3-manifolds (1997)
[8] DOI: 10.1080/10586458.1999.10504390 · Zbl 0927.57020 · doi:10.1080/10586458.1999.10504390
[9] Thurston W. P., Three-dimensional geometry and topology 1 (1997) · Zbl 0873.57001
[10] DOI: 10.2140/pjm.1998.183.359 · Zbl 0930.57017 · doi:10.2140/pjm.1998.183.359
[11] Tollefson J. L., Topology 35 (1) pp 55– (1996) · Zbl 0868.57022 · doi:10.1016/0040-9383(95)00008-9
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