After this paper was published, the following information about doodles was pointed out by Roger Fenn. A doodle was introduced by Fenn and Taylor , which is a finite collection of closed curves without triple intersections on a closed oriented surface considered up to the second flat Reidemeister moves with the condition (*) that each component has no self-intersections. Khovanov  introduced doodle groups, and for his process, he considered doodles under a more generalized setting (i.e. removing the condition (*) and permitting the first flat Reidemeister moves). He showed [4, Theorem 2.2], a result similar to our [3, Theorem 2.2(c)]. He also pointed out that [1, Corollary 2.8.9] gives a result similar to [4, Theorem 2.2].
The authors first noticed the above results by Fenn and Khovanov via personal communication with Fenn, and therefore, the authors would like to thank Roger Fenn for these references.
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