Maximal TB number and slice genus relation of a knot in any 3-manifold Lisca-Stipsicz say that if a knot $K\subset{S^{3}}$ satisfies $g_{s}(K)>{0}$ and                  $TB(K)=2g_{s}(K)-1$ then $S^{3}_r(K)$ carries positive, tight contact structures
for every $r\neq{TB(K)}$ where $S^{3}_r(K)$ is the new manifold obtained by performing $r$-surgery to $K$ in $S^{3}$. Can we say this result is true for any 3-manifold not only for $S^{3}$?
 A: If $K \subset (M,\xi)$ is null-homologous, then it makes sense to define $TB(K)$ as in $S^3$.  The right definition of $g_s(K)$ is as a minimum over the genera of smooth surfaces in $M \times [0,1]$ with boundary $K \times \{1\}$.
With a null-homologous knot satisfying $TB(K) = 2g_s(K) -1$, and when the Heegaard Floer contact invariant $c(\xi)$ is non-vanishing, then $M_r(K)$ has a tight contact structure for all $r \neq TB(K)$.  For $r < TB(K)$, this follows from the fact that Legendrian surgery on any link preserves the non-vanishing of the Heegaard Floer contact invariant (in particular, choose the link as in Ding and Geiges's Surgery diagrams for contact 3-manifolds); for $r > TB(K)$, this result can be found as Theorem 1.7 in my pre-print Transverse Surgery on Knots in Contact 3-Manifolds.
When $c(\xi) = 0$ but $\xi$ is still tight, I believe that we know nothing except in the case where $K$ is fibred, whereupon $M_r(K)$ supports a tight contact structure for all $r \neq TB(K)$ (this is Theorem 1.6 in the above pre-print).
