Semimetals without correspondence between the topological charge of nodal line/surface and Fermi arc

It is generally believed that there is a correspondence between the topological charge of nodal points or lines and the presence of Fermi arcs. Using a $\mathcal{P}\mathcal{T}$-invariant system as an example, we demonstrate that this general belief is no longer valid. When two charged nodal lines or surfaces touch, the topological charge dissipates without gap opening, yet the surfaces or hinge Fermi arcs can remain preserved. It is found that both static and Floquet semimetals can exhibit Fermi arcs, even when the nodal lines or surfaces do not carry a ${Z}_{2}$ charge from the second Stiefel-Whitney class.