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Deforming abelian SU(2)-representations of knot groups. (English) Zbl 0910.57004

Let \(k\subset S^3\) be a knot and \(G=\pi_1(S^3\smallsetminus k)\) be its group. Let \(m \in G\) be a meridian and let \(\alpha \in [0,\pi]\) be a given real number. Define an abelian representation \(\rho_\alpha : G \to SU(2)\) by \(\rho_\alpha(m)=\left(\begin{smallmatrix} e^{i \alpha}&0\\ 0&e^{i \alpha}\end{smallmatrix}\right).\) Denote the Alexander polynomial of \(k\) by \(\Delta_k\). It was shown by E. P. Klassen [Trans. Am. Math. Soc. 326, No. 2, 795-828 (1991; Zbl 0743.57003)] that if \(\rho_\alpha\) is a limit of non-abelian representations \(\rho : G \to SU(2)\) then \(\Delta_k(e^{2i\alpha})=0\). It is conjectured that these condition is also sufficient. The conjecture under assumption that \(e^{2i\alpha}\) is a single root of \(\Delta_k\) was proven by C. D. Frohman and E. P. Klassen [Comment. Math. Helv. 66, No. 3, 340-361 (1991; Zbl 0738.57001)].
In the present paper the generalization of this result is given. The main theorem of the article states that if \(\Delta_k(e^{2i\alpha})=0\) and the signature function \(\sigma_k : S^1 \to Z\) changes its value at \(e^{2i\alpha}\) then the abelian representation \(\rho_\alpha\) is an endpoint of an arc of non-abelian representations. Remark that \(\sigma_k\) changes its value at \(\omega \in S^1\) if \(\omega\) is a root of \(\Delta_k\) of odd multiplicity. The proof makes use of a generalization of a result of X.-S. Lin [J. Differ. Geom. 35, No. 2, 337-357 (1992; Zbl 0774.57007)]. Independently this result was proven by C. M. Herald [Math. Ann. 309, No. 1, 21-35 (1997; Zbl 0887.57013)].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
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