We discuss the dynamics of an inextensible thin Kirchhoff rod used in the modeling of surgical threads, and demonstrate a very efficient scheme to not only simulate the motion of the thread in real-time (up to 1 ms per frame) but also obtain the constraint axial forces which can be fed back to a haptic system. The numerical scheme is based on a family of schemes called geometric or discrete variational integrators guaranteeing that the momentum and energy are exactly conserved over long periods of time for conservative systems. Besides, we report on an efficient numerical procedure to handle the inextensibility of the thread through physically based Lagrange multipliers, as well as the internal dissipation of the thread. We have performed simulations to verify the capabilities of our model to conserve momentum and energy, accurately calculate the axial constraint forces along the thread for haptic feedback, and capture bending-torsion coupling leading to the formation of plectonemes. While many of the ideas are well known in the computer graphics community (especially in hair modeling), we have implemented several improvements for the specific purpose of speeding up the computations for developing physically based haptic interfaces for knot tying and suturing.