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The \(SL(3,\mathbb{C})\)-character variety of the figure eight knot. (English) Zbl 1373.57014

The character variety of a three-manifold group in \(\mathrm{SL}(2, \mathbb C)\) is a useful tool in the study of the geometric and topological properties of the three-manifold. Much less is known for the character varieties of three-manifold groups in other Lie groups, notably for \(\mathrm{SL}(r,\mathbb C)\), \(r\geq 3\).
In this paper, the authors give explicit equations that describe the character variety of the figure eight knot for the groups \(\mathrm{SL}(3, \mathbb C)\), \(\mathrm{GL}(3, \mathbb C)\) and \(\mathrm{PGL}(3, \mathbb C)\). They also describe each of the five components of the character variety obtained in the three cases and study the action of the symmetry group of the figure eight knot on the character varieties.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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Full Text: arXiv Euclid