I am trying to understand the algebraic intersection number in terms of Poincare dual and the cup product. This is:
Let $M$ be a compact oriented $m$-dimensional smooth manifold together with a boundary decomposition $\partial M = A\cup B$. Let $X$ and $Y$ be a complementary pair of submanifolds of $M$ with $\partial X\subset A$ and $\partial Y\subset B$. We denote by $i:X\rightarrow M$ and $j:Y\rightarrow M$ the obvious inclusion maps. Furthermore we denote by $\left[ X\right] \in H_k(X,\partial X)$ and $\left[ Y\right] \in H_{m-k}(Y,\partial Y)$ the fundamental classes of $X$ and $Y$. Finally denote by
$$PD_M:H_{m-k}(M,B)\rightarrow H^{k}(M,A)$$
the inverse of Poincare duality.
Then
$$X\cdot Y= \langle \text{PD}_M(i_\star(\left[ X\right] ))\cup \text{PD}_M(j_\star(\left[ Y\right] )),\left[ M\right] \rangle.$$
Now let $K,J\subset S^3$ be knots and let $F_K$ be some Seifert-surface of $K$ such that $F_K$ and $J$ are transverse. Then for $T:B^2\times K\rightarrow S^3$ an appropiate thickening and $M=X_K=S^3\setminus T(B^2\times K)$ we obtain
$$L\cdot F_K = \langle \underbrace{\text{PD}_{X_K}(j_\star(\left[ L\right] ))}_{\color{red}{=??}}\cup \text{PD}_{X_K}(i_\star(\left[ F^\prime\right] )),\left[ X_K\right] \rangle$$
As highlighted in red I want to figure out what the Poincare dual of $j_\star ([L])$ is.
Any help would be appreciated!