## 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

### Citations:

Zbl 0695.57011; Zbl 0774.57007; Zbl 0339.55005
Full Text:

### References:

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