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A volume form on the SU(2)-representation space of knot groups. (English) Zbl 1095.57012

Let \({k} \subset S^3\) denote a knot. The author studies the space \(\mathcal{R}eg({k})\) of conjugacy classes of “regular” irreducible maps from \(\pi_1(S^3-{k})\) to SU(2).
Adapting ideas of A. Casson [described in S. Akbulut and J. D. McCarthy, Casson’s invariant for oriented homology 3-spheres, Mathematical Notes, 36. Princeton University Press (1990; Zbl 0695.57011) and X.-S. Lin, J. Differ. Geom. 35, 337–357 (1992; Zbl 0774.57007)], as well as M. Heusener and E. Klassen, he defines a 1-form \(\omega_{k}\) on the 1-dimensional manifold \(\mathcal{R}eg({k})\). Using work of J. Birman [Can. J. Math. 28, 264–290 (1976; Zbl 0339.55005)], he shows that \(\omega_{k}\) is a knot invariant.
He calculates \(\omega_{k}\) for the case of a \((2,q)\) torus knot. He also proves the integral of the invariant over \(\mathcal{R}eg({k})\) is additive with respect to the addition of knots.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20C99 Representation theory of groups
58A10 Differential forms in global analysis
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References:

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