Why a regular neighborhood of a connected, compact, orientable surface F with boundary in $S^3$ is a handlebody of genus 2g+n-1? (where g is a gunus of F, and n is the number of boundary components of F.)
(This is mentioned without proof in Lickokish's book "An Introduction to Knot Theory"( in the proof of Proposition 6.3.) .)
Best Answer
Because $F$ has a deformation retract $G \subset F$ which is a rose with $2g+n-1$ petals, and any regular neighborhood of $F$ is also a regular neighborhood of $G$, which is a handlebody of genus $2g+n-1$.