Metabelian SL(n,C) representations of knot groups

Details Metabelian SL(n,C) representations of knot groups Metabelian SL(n,C) representations of knot groups

2008-03-30

arxiv.org/abs/0803.4329v1

Pacific J. Math. Vol. 238 (2008), No. 1, 7-25

Mathematics

Geometric Topology

18 pages

Scientific paper

We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.

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