Topologija za krojače (ali kako ploskev razrezati na hlače)

V tem članku bo predstavljeno, kako lahko nekatere orientabilne ploskve razrežemo na komponente, homeomorfne sferi s tremi luknjami (oziroma hlačam). Ključno vlogo bodo imeli rezni sistemi, ki povedo, kako razrezati ploskev. Nato bodo predstavljene dekompozicije ploskev. To so množice komponent, na katere rezni sistem razreže ploskev. Nekaj več bo povedanega o hlačah. To so ploskve, homeomorfne sferi s tremi luknjami. Podrobneje bo predstavljen poseben primer dekompozicij – hlačne dekompozicije. To so dekompozicije, pri katerih je vsaka komponenta homeomorfna hlačam. Povedano bo, za katere orientabilne ploskve hlačne dekompozicije sploh obstajajo. Nato bo pokazano, kako lahko hlačni dekompoziciji priredimo multigraf, oziroma kako hlačno dekompozicijo „zakodirati“ z multigrafom. Nato bodo predstavljeni premiki. To so načini za prehajanje med različnimi hlačnimi dekompozicijami dane ploskve. Na koncu bosta dokazana ključna rezultata, ki zagotovita, da so premiki dobro definirani, tj. premiki iz hlačne dekompozicije vedno dajo novo hlačno dekompozicijo.

How to cut a surface into pants

In this article, it will be shown how some orientable surfaces can be cut into components, homeomorphic to a sphere with three holes (also known as a pair of pants). Cuttings of surfaces will be discussed. Cut systems will have a crucial role here. Cut systems tell us how to cut the surface. Then the decompositions of surfaces will be considered. Those are sets of components into which the cut system cuts our surface. A surface, known as pants, will be presented next. Those are surfaces homeomorphic to a sphere with three holes. Then, the focus will be shifted to a special type of decomposition, namely pants decompositions. Those are decompositions, where every component is homeomorphic to a pair of pants. It will be shown for which orientable surfaces pants decompositions exist. Then, a method for assigning a multigraph to a pants decomposition will be presented. Conversely, it will be shown how a pants decomposition can be “encoded” with a multigraph. Then the moves will be presented. Those are ways to move between different pants decompositions of the given surface. In the end, two essential results that ensure the well-definedness of moves will be proven, i.e., the moves always give us a new pants decomposition.