# Is there an analog of Reidemeister's theorem for braids in a surface?

Reidemeister's classical theorem describes the set of links in $\mathbb R^3$ up to isotopy as the set $\{ \textrm{diagrams in } \mathbb R^2\textrm{ with crossings}\}$ modulo certain local relations on diagrams, each of which obviously preserves the isotopy type of the link that the diagrams represent.

Given a surface $F$ with $n$ marked points $\{p_i\} \subset F$, one can consider its braid group $Br(F,n)$, where a braid is a set of $n$ oriented embedded arcs in $F\times [0,1]$, which start at $\{p_i\}\times 0$, end at $\{p_i\}\times 1$, and are always increasing in the $[0,1]$ direction. Then "braids up to isotopy" is a group, with composition given by "stacking in the $[0,1]$ direction." If $F$ is a disc, we get the classical braid group.

We can represent a braid by a "diagram with crossings" on the surface $F$, where the diagrams consist of oriented arcs which begin and end at one of the marked points on $F$.

Q: Are there "Reidemeister moves" for diagrams representing surface braids, such that quotienting diagrams by these moves gives you the braid group $Br(F,n)$?

• Transactions of the AMS 356 (1), 2004, 219-243 by Juan González-Meneses and Luis Paris would be a good place to start. It's not the reference you want, but I think they cite the reference you want, if I recall correctly. – Ryan Budney Dec 14 '16 at 16:33
• Another good reference is Bellingeri " On presentation of Surface Braid Groups" arxiv.org/abs/math/0110129. – Adrien Dec 14 '16 at 16:42