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Homological mirror symmetry predicts that there is a relation betweenautoequivalence groups of derived categories of coherent sheaves on Calabi-Yauvarieties, and the symplectic mapping class groups of symplectic manifolds. Inthis paper, as an analogue of Dehn twists for closed oriented real surfaces, westudy spherical twists for dg-enhanced triangulated categories.  We introduce the intersection number and relate it to group-theoreticproperties of spherical twists. We show an inequality analogous to afundamental inequality in the theory of mapping class groups about the behaviorof the intersection number via iterations of Dehn twists. We also classify thesubgroups generated by two spherical twists using the intersection number. Inpassing, we prove a structure theorem for finite dimensional dg-modules overthe graded dual numbers and use this to describe the autoequivalence group.  As an application, we compute the center of autoequivalence groups of derivedcategories of K3 surfaces.

Spherical twists, relations and the center of autoequivalence groups of K3 surfaces