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Swirling flows abound in nature and numerous engineering applications. Under conditions which are not completely understood, the swirling cores could undergo a sudden enlargement of their vortex core, leading to a ’vortex breakdown’. The physics of vortex breakdown and strategies to control it have been active areas of research for nearly half a century. There are many competing theories of vortex breakdown in the literature; broadly, these are surmised on similarities to flow separation, hydrodynamic instability or transition from a supercritical to a subcritical state. However, a rational criterion for vortex breakdown continues to be elusive. One of the most well known criteria in the literature is the one due to Brown and Lopez (1990) based on an inviscid vortex dynamics model which suggests that the helix angle of the velocity vector should enclose the helix angle of the vorticity vector. However it appears that this only suggests that the stream surface would diverge and not necessarily constitute a condition for breakdown. In this work, we propose a new criterion based on helicity (scalar product of velocity and vorticity vectors) for characterizing breakdown since it has fundamental topological interpretations relating to change in linkages of vortex lines. In particular, it is suggested that the breakdown location corresponds to the location where helicity becomes zero. We study the problem of vortex breakdown in a cylindrical container with a rotating top lid in order to clarify and elucidate our hypothesis. We present results from Direct Numerical Simulation of this problem for three different Reynolds numbers and evaluate the utility of our proposed helicity criterion. Our studies indicate that helicity is indeed a better choice for characterizing vortex breakdown.