Abstract
We consider a groupoid structure involving graphs and imbedded circles (resp. disks) in a closed orientable surface F (resp. the handleboby H it bounds). We look at generalized connected sums and related operations "induced" by certain morphisms, and introduce a quandle structure involving isotopy classes of imbedded circles and Dehn twists in F. Some relations between these operations are examined, and we look at characterizations of imbedded disks in H arising in this algebraic context.
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