  In this paper, we provide a complete classification of the positive minimal
monads whose cohomology is a stable rank 2 bundle on $\mathbb{P}^3$ with Chern
classes $c_1=-1, c_2=10$ and we prove the existence of a new irreducible
component of the moduli space $\mathcal{B}(-1,10)$ of a rank 2 stable bundles
with the given Chern classes. We also show that Hartshorne's conditions on a
sequence $\mathcal{X}$ of 10 integers are sufficient and necessary for the
existence of a stable rank 2 bundle with odd determinant and spectrum
$\mathcal{X}$. Furthermore, we prove that the sequence of integers
$\{-2^{n-1},-1,0,1^{n-1}\}$ for $ n\geq4$ is realized as the spectrum of a
stable rank 2 bundle $\EE$ of odd determinant by computing the minimal
generators of its Rao module.
