Looking for a straightforward proof that the negative Hopf Band, $H^-$ induces an overtwisted contact structure on $S^3$. I'm aware that this can be done by showing that the Thurston-Bennequin inequality for Legandrian knots is violated (see page 18 of https://arxiv.org/pdf/math/0111118.pdf for details).
Is there a way that this can be shown directly? I.e., by explicitly computing the self-linking number and rotation number?
Alternatively, if this is not possible, is there a way to simply construct an overtwisted disk in the contact structure? I'm having trouble explicitly constructing what the contact structure is (the one form which is associated with it).
Any help is appreciated!