A hyperlink is a finite set of non-intersecting simple closed curves in $$\mathbb {R} \times \mathbb {R}^3$$. Let R be a compact set inside $$\mathbb {R}^3$$. The dynamical variables in General Relativity are the vierbein e and a $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$-valued connection $$\omega $$. Together with Minkowski metric, e will define a metric g on the manifold. Denote $$V_R(e)$$ as the volume of R, for a given choice of e. The Einstein–Hilbert action $$S(e,\omega )$$ is defined on e and $$\omega $$. We will quantize the volume of R by integrating $$V_R(e)$$ against a holonomy operator of a hyperlink L, disjoint from R, and the exponential of the Einstein–Hilbert action, over the space of vierbein e and $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$-valued connection $$\omega $$. Using our earlier work done on Chern–Simons path integrals in $$\mathbb {R}^3$$, we will write this infinite-dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the volume operator can be computed by counting the number of nodes on the projected hyperlink in $$\mathbb {R}^3$$, which lie inside the interior of R. By assigning an irreducible representation of $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$ to each component of L, the volume operator gives the total kinetic energy, which comes from translational and angular momentum.