  A nodal-line semimetal is a topological gapless phase containing
one-dimensional degeneracies called nodal lines. The nodal lines are deformed
by a continuous change of the system such as pressure and they can even change
their topology, but it is not systematically understood what kind of changes of
topology of nodal lines are possible. In this paper, we classify the events of
topology change of nodal lines by the Morse theory and reveal that only three
types of topology changes of nodal lines, i.e., creation, reconnection, and
annihilation, are possible in the spinless nodal-line semimetal protected by
inversion and time-reversal symmetries. They are characterized by an index
having the values 0, 1, and 2 for the above three types in the Morse theory.
Moreover, we extend our theory to systems with rotational symmetries and mirror
symmetry and disclose the possible events of topology change of nodal lines
under each symmetry.
