Let $\Sigma$ be a genus $g$ closed orientable surface and let $\alpha = \{ \alpha_1,...,\alpha_g\}$ and $\beta = \{ \beta_1 ,..., \beta_g \}$ be two cut systems (i.e. nonintersecting homologically linearly independent sets of $g$ curves) on $\Sigma$. Is there an algorithm to tell if $\alpha$ is handleslide equivalent to $\beta$? Or said another way, is there an algorithm to tell if $\alpha$ and $\beta$ describe the same handlebody attached to $\Sigma$?
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Yes, here is a sketch of one such proof. Let $V = V_\alpha$ be the genus $g$ handlebody determined by $\alpha$. The curves $\beta$ now give conjugacy classes in $\pi_1(V)$. Any one of the $\beta_i$ bounds a disk if and only if its class is trivial. (Strictly speaking, this requires the disk theorem, but there is a easy proof when the ambient three-manifold is a handlebody.) If all of the $\beta_i$ bound disks in $V$, then an innermost disk argument proves that the $\beta_i$ bound disjointly. Now an outermost bigon argument gives the desired sequence of handleslides from $\beta$ to $\alpha$.
There are also indirect proofs. See Theorem 1.4 of a paper of Carvalho and Oertel or Lemma 2.2 of a paper of Casson and Long.
