Two geometric models for graded skew-gentle algebras

In Part 1, we classify (indecomposable) objects in the perfect derived category $\mathrm{per}\Lambda$ of a graded skew-gentle algebra $\Lambda$, generalizing technique/results of Burban-Drozd and Deng to the graded setting. We also use the usual punctured marked surface $\mathbf{S}^\lambda$ with grading (and a full formal arc system) to give a geometric model for this classification. In Part2, we introduce a new surface $\mathbf{S}^\lambda_*$ with binaries from $\mathbf{S}^\lambda$ by replacing each puncture $P$ by a boundary component $*_P$ (called a binary) with one marked point, and composing an equivalent relation $D_{*_P}^2=\mathrm{id}$, where $D_{*_p}$ is the Dehn twist along $*_P$. Certain indecomposable objects in $\mathrm{per}\Lambda$ can be also classified by graded unknotted arcs on $\mathbf{S}^\lambda_*$. Moreover, using this new geometric model, we show that the intersections between any two unknotted arcs provide a basis of the morphisms between the corresponding arc objects, i.e. formula $\mathrm{Int}=\mathrm{dim}\mathrm{Hom}$ holds.