I am reading Birman and Brendle's 2004 article, Braids: A Survey. In a section introducing the Yamada-Vogel algorithm, the authors use the notation $H_1(A)$. I did some research and I think the $H_1$ may refer to a handlebody of index $1$. $H_1(A)$ is apparently a set? If that's the case, then I don't understand this notation. I've attached the page for context.
Birman & Brendle, Braids: A Survey, pg. 13
And here are the sentences I need to understand: "Let $C$ and $C'$ be two oriented disjoint simple closed curves in $S^2$. Then $C$ and $C'$ cobound an annulus $A$. We say that $C$ and $C'$ are coherent (or coherently oriented) if $C$ and $C'$ represent the same element of $H_1(A)$."
$H_1(A)$ is the first (singular) homology group of the annulus $A$. Homology is a homotopy invariant, and since an annulus deformation retracts onto one of the base circles $S^1$, we have $$H_1(A) \cong H_1(S^1) \cong \mathbb{Z}.$$
An element of this group may be understood as a winding number around the circle. A choice of generator of this group simply corresponds to an orientation of the circle, i.e., a full loop around the circle in either direction. Therefore, the two base circles are coherently oriented iff they represent the same generator in this group.