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In this article, we generalize the results discussed in [arXiv:1004.3762] byintroducing a genus to generic fibers of Lefschetz fibrations. That is, we givefamilies of relations in the mapping class groups of genus-1 surfaces withboundaries that represent rational homology disk smoothings of weightedhomogeneous surface singularities whose resolution graphs are $3$-legged with abad central vertex.

Rational homology disk smoothings and Lefschetz fibrations