  Checkerboard framings are an extension of checkerboard colorings for virtual
links. According to checkerboard framings, in 2017, Dye obtained an independent
invariant of virtual links: the cut point number. Checkerboard framings and cut
points can be used as a tool to extend other classical invariants to virtual
links. We prove that one of the conjectures in Dye's paper is correct.
Moreover, we analyze the connection and difference between checkerboard framing
obtained from virtual link diagram by adapting cut points and twisted link
diagram obtained from virtual link diagram by introducing bars. By adjusting
the normalized arrow polynomial of virtual links, we generalize it to twisted
links. And we show that it is an invariant for twisted link. Finally, we figure
out three characteristics of the normalized arrow polynomial of a checkerboard
colorable twisted link, which is a tool of detecting checkerboard colorability
of a twisted link. The latter two characteristics are the same as in the case
of checkerboard colorable virtual link diagram.
