Whitney circle for a pair of intersection points

Let $X^m$ be a smooth manifold without boundary and let $A^a,B^b\subseteq X$ be closed embedded submanifolds meeting transversely with $a+b=m$ (Smooth manifolds and their smooth charts, Embedded submanifolds and slice charts, Transverse embedded submanifolds). For two distinct transverse intersection points $p,q\in A\cap B$ a Whitney circle for the ordered pair $(p,q)$ is a closed curve $\gamma =\alpha \ast \beta$ in $X$, where $\alpha :I\to A$ is a smooth embedded arc from $p$ to $q$, $\beta :I\to B$ is a smooth embedded arc from $q$ to $p$, and both arcs meet $A\cap B$ only in their endpoints: $\alpha ((0,1))\cap (A\cap B)=\mathrm{\varnothing }$ and $\beta ((0,1))\cap (A\cap B)=\mathrm{\varnothing }$, with $\alpha$ and $\beta$ otherwise disjoint. The arcs are part of the data, not determined by the pair $(p,q)$: different arcs give loops that differ by loops in $A$ and in $B$. Equivalently, $\gamma :S^1\to X$ is an embedding, smooth except at the corners $p,q$, whose two branches lie in $A$ and in $B$ respectively, meet $A\cap B$ only at $p,q$, and have linearly independent tangent directions there. No orientation, coefficient system or dimension inequality beyond $a+b=m$ is imposed, and the definition asserts nothing about existence of such arcs.

Here $I=[0,1]$ is the closed interval, an arc is a smooth embedding of $I$ (Smooth embeddings), and the concatenation $\alpha \ast \beta$ traverses $\alpha$ and then $\beta$, so $\gamma (0)=\gamma (1)=p$. The condition $a+b=m$ is exactly complementary dimension: it is what makes $T_{p}X=T_{p}A\oplus T_{p}B$ at each transverse intersection point, so that the two branch tangent lines are linearly independent at the corner. The two arcs of a Whitney circle avoid every double point of $A\cap B$ other than $p$ and $q$, which is what a later clean Whitney disk must span. The definition is a naming of the boundary object only: it imposes no orientability, no coefficient system, no inequality between $a$ and $b$ beyond $a+b=m$, and it neither asserts nor denies that arcs with these properties exist; the arcs lemma on this page supplies them for connected sheets of dimension at least two.