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We study a generalized version of the Hamiltonian constraint operator innonperturbative loop quantum gravity. The generalization is based on admittingarbitrary irreducible SU(2) representations in the regularization of theoperator, in contrast to the original definition where only the fundamentalrepresentation is taken. This leads to a quantization ambiguity and to a familyof operators with the same classical limit. We calculate the action of theEuclidean part of the generalized Hamiltonian constraint on trivalent states,using the graphical notation of Temperley-Lieb recoupling theory. We discussthe relation between this generalization of the Hamiltonian constraint andcrossing symmetry.Comment: 35 pp, 20 eps figures; minor corrections, references added; version  to appear in Class. Quant. Gra

A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean matrix elements