Zero-one Grothendieck polynomials

Fink et al. (2020)showed that the Schubert polynomial $~\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial $~\mathfrak{G}_w(x)$ is zero-one, i.e., with coefficients either 0 or $\pm$1, if and only if $w$ avoids six patterns. As applications, we show that the normalized double Schubert polynomial $N(~\mathfrak{S}_w(x;y))$ is Lorentzian when $~\mathfrak{G}_w(x)$ is zero-one, partially confirming a conjecture of Huh et al. (2022). Moreover, we verify several conjectures on the support and coefficients of Grothendieck polynomials posed by Mészáros et al. (2025) for the case of zero-one Grothendieck polynomials.