Trisections of surface complements and the Price twist

Given an $S\cong \mathbb{R}{P}^2$ smoothly embedded in a 4-manifold $X^4$ with Euler number 2 or -2, the Price twist is a surgery operation on $\nu (S)$ yielding (up to) three different 4-manifolds: $X^4,{\tau }_S(X^4),{\mathrm{\Sigma }}_S(X^4)$. This is of particular interest when $X^4=S^4$, as then ${\mathrm{\Sigma }}_S(X^4)$ is a homotopy 4-sphere which is not obviously diffeomorphic to $S^4$. In this paper, we show how to produce a trisection description of each Price twist on $S\subset X^4$ by producing a relative trisection of $X^4\setminus \nu (S)$. Moreover, we show how to produce a trisection description of general surface complements in 4-manifolds.