Two geometric models for graded skew-gentle algebras

In Part 1, we classify (indecomposable) objects in the perfect derived category $\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{\Lambda }$ of a graded skew-gentle algebra $\mathrm{\Lambda }$, 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 ${\mathbf{S}}_{\ast }^{\lambda }$ with binaries from ${\mathbf{S}}^{\lambda }$ by replacing each puncture $P$ by a boundary component ${\ast }_P$ (called a binary) with one marked point, and composing an equivalent relation $D_{{\ast }_P}^2=\mathrm{i}\mathrm{d}$, where $D_{{\ast }_p}$ is the Dehn twist along ${\ast }_P$. Certain indecomposable objects in $\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{\Lambda }$ can be also classified by graded unknotted arcs on ${\mathbf{S}}_{\ast }^{\lambda }$. 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 $\mathrm{I}\mathrm{n}\mathrm{t}=\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}\mathrm{o}\mathrm{m}$ holds.