  We present a framework for studying transverse knots and symplectic surfaces
utilizing the Seiberg-Witten monopole equation. Our primary approach involves
investigating an equivariant Seiberg-Witten theory introduced by
Baraglia-Hekmati on branched covers, incorporating invariant contact/symplectic
structures.
  Within this framework, we introduce a novel slice-torus invariant denoted as
$q_M(K)$. This invariant can be viewed as the Seiberg-Witten analog of
Hendricks-Lipshitz-Sarker's $q_\tau$ invariant, with a signature correction
term. One property of the invariant $q_M(K)$ is an adjunction equality for
properly embedded connected symplectic surfaces in the symplectic filling
$D^4\# m \overline{\mathbb{C}P}^2$. The proof of this equality utilizes the
equivariant version of the homotopical contact invariant introduced by the
authors, leading to a transverse knot invariant. Another ingredient of the
proof involves constructing invariant symplectic structures on branched
covering spaces branched along properly embedded symplectic surfaces in
symplectic fillings. As an application of the invariant $q_M(K)$, we determine
the value of any slice-torus invariant within a permissible deviation of $2$
for squeezed knots concordant to certain Montesinos knots. Additionally, we
provide an obstruction to realizing second homology classes of $D^4 \#m
\overline{\mathbb{C}P}^2$ as connected embedded symplectic surfaces with
transverse knot boundary or connected embedded Lagrangian surfaces with
collarable Legendrian knot boundary. Moreover, we introduce a new obstruction
to certain Montesinos knots being quasipositive, which is described only in
terms of slice genera and their signatures.
