# “Basic” loops on standardly embedded surfaces

Take a genus $$g$$ surface $$S$$ standardly embedded in $$\mathbb{R}^3$$, by which I mean it is unknotted. Surface $$S$$ bounds a volume $$V$$ that deformation retracts on a standardly embedded planar graph $$G$$ with $$\beta_1 = g$$, and that only has degree $$3$$ vertices.

Among the loops on $$S$$ that are null homotopic in $$V$$, there is a subset that are boundaries of embedded disks in $$V$$ that intersect $$G$$ exactly once for some choice of $$G$$ as above.

Do these loops (or perhaps close variants) have a name? Do they have an alternate definition?

They are often called meridians of $$G$$. Note that there are many graphs $$G$$ to which $$V$$ deformation retracts (most nonplanar); if you are not particular about which graph $$G$$ then they are called meridians of $$V$$. $$V$$ is called a handlebody and $$G$$ is a spine of the handlebody. See, for example, Scharlemann's "Refilling meridians in a genus 2 handlebody complement" arXiv:math/0603705
• They are essentially equivalent ideas. Suppose $D$ is a collection of essential discs in $V$ such that no two are isotopic and such that any other essential disc disjoint from $D$ is isotopic to a disc in $D$. Then cutting $V$ along $D$ produces a collection of 3-balls with 3 scars from $D$ in each boundary. Cone the center of each scar to the center of the 3-ball to get a "tripod". Undoing the cutting along $D$ connects the tripods together (or two edges of a tripod together) to form a graph $G$ to which $V$ deformation retracts. – Scott Taylor May 30 '20 at 20:42
• Also the graph $G$ and the discs $D$ satisfy your requirements, except that $G$ may not be planar, and given any embedded disc in $V$ with boundary essential in $S$ it is contained in such a collection of discs. (That is little harder to prove, but is a standard innermost circle/outermost arc argument. Probably you can find the argument in Jaco's or Hempel's books on 3-manifolds.) Insisting that $G$ be planar is much more restrictive and the corresponding discs are really associated more to $G$ than to $V$. I would still call them meridian discs though. – Scott Taylor May 30 '20 at 20:50