Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. Turbulence can thus be understood as a walk along dynamical connections between those solutions. In particular the current accepted view is that turbulent trajectory is transiently attracted to an individual invariant solution along its stable manifold before being ejected along the unstable manifold and captured by another one. Thus, to characterize the turbulence-supporting saddle we not only need to compute the invariant states but also identify connections between them. In this talk, we first investigate the presence of heteroclinic connections as a natural path connecting invariant solutions, and followed by turbulent trajectories during the time evolution. Thus, we propose a robust variational method for computing heteroclinic orbits between equilibrium solutions. By using it, we find previously unknown connections capturing various aspects of the turbulent dynamics including characteristic bursting events. Although this scenario is well established, we investigate the possibility that on a finite time horizon trajectories may also leave exact invariant solutions along an asymptotically attracting direction in the stable manifold due to the non-normality of the Navier-Stokes equations. In particular, we look for optimally amplified trajectories leaving the neighbourhood of invariant solutions. We show that well known vortical structures, i.e. hairpin vortices, commonly observed in wall bounded shear flows, emerge from the exact invariant solutions when perturbed in specific directions in its stable manifold, bypassing the unstable one. These results not only provide a new mechanism by which hairpin vortices emerge from invariant solutions, but also challenge the common picture of a turbulent trajectory chaotically wandering between invariant solutions leaving them along their unstable manifold before being attracted to another others along their stable manifold. This counter-intuitive result therefore suggests that our understanding of the geometry of turbulence in wall bounded flows might have been incomplete and that stable manifolds seem extremely important to understand how the network of unstable solutions supporting turbulence is formed.