  Arkani-Hamed, Bai, He, and Yan (ABHY) discovered a convex realisation of the
associahedron whose combinatorial and geometric structure generates tree-level
amplitudes in bi-adjoint scalar theory. In this paper, we identify S-matrix of
Yang-Mills theory with a scalar obtained by contracting the canonical form of
ABHY associahedron with a multi-vector field (MVF) in the kinematic space.
Components of this MVF are determined by the combinatorial structures that
underlie the associahedron and Corolla polynomial that was introduced by
Kreimer, Sars, and van Suijlekom (KSVS) in [2]. KSVS used the Corolla
polynomial to obtain (at all orders in the loop expansion) the parametric
representation of gauge theory Feynman integral from the corresponding Feynman
integral in $\phi^{3}$ theory. Using the full power of Corolla polynomial, we
then extend these results to obtain Yang-Mills one loop planar integrand by
contracting the Corolla generated MVF with the canonical form defined by
$\hat{D}_{n}$ polytope discovered by Arkani-Hamed, Frost, Plamondon, Salvatori,
Thomas. We also demonstrate that KSVS representation of Corolla graph
differential in the parametric space can be readily extended to "spin up" the
curve integral formulae for $\textrm{Tr}\phi^{3}$ amplitude discovered in [3,4]
and give an explicit construction of such formulae for tree-level and planar
one loop gluon amplitudes.
