  Mirror symmetry for a toric variety involves Laurent polynomials whose
symplectic topology is related to the algebraic geometry of the toric variety.
We show that there is a monodromy action on the monomially admissible
Fukaya-Seidel categories of these Laurent polynomials as the arguments of their
coefficients vary that corresponds under homological mirror symmetry to
tensoring by a line bundle naturally associated to the monomials whose
coefficients are rotated. In the process, we introduce the monomially
admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel
category of a Laurent polynomial on $(\mathbb{C}^*)^n$, which has other
potential applications, and give evidence of homological mirror symmetry for
non-compact toric varieties.
