The fork-noodle pairing detects essential intersections

Assume first that $T(F)$ is isotopic relative to $\mathrm{\partial }D\cup P$ to an arc disjoint from $N$. Choose such a representative $F^{\mathrm{\prime }}$ of the isotopy class of $F$ with $T(F^{\mathrm{\prime }})\cap N=\mathrm{\varnothing }$; the finitely many intersection points of $T(F^{\mathrm{\prime }})$ with $N$ number $l=0$. Then the geometric sum of [F1] is empty, so $⟨N,F^{\mathrm{\prime }}⟩=0$, and representative-independence in [F1] gives $⟨N,F⟩=0$. This proves the implication from disjointness to vanishing.

For the converse, suppose $T(F)$ cannot be isotoped relative to $\mathrm{\partial }D\cup P$ to an arc disjoint from $N$. By [F3] every minimal-position representative of $T(F)$ meets $N$; choose such a representative $T^{\ast }$ and let $l\ge 1$ be its number of intersection points with $N$; the corresponding fork is isotopic to $F$ relative to $\mathrm{\partial }D\cup P$. The minimal position hypothesis of [F2] is satisfied, so the maximal monomial of the geometric sum occurs with a single sign and nonzero coefficient, whence $⟨N,F^{\ast }⟩\mathrm{\ne }0$. By representative-independence in [F1] again, $⟨N,F⟩=⟨N,F^{\ast }⟩\mathrm{\ne }0$. This proves the contrapositive.