Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tor...Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tori X, Y with A belong X, B belong Y. Let γ be a simple closed curve on F which is disjoint from A ∪ B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hγ^n（C） ： n ∈ Z} is the set containing all the simple closed curves on F which is separating with respect to （A, B）, where hγ is the Dehn twist along γ on F. This also shows how two simple closed curves on F which are separating with respect to （A, B） are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.展开更多
基金The NSF (10171024,10471020) of Chinaa grant of the outstanding Youth fellowship of Hei Long Jiang Province and the Science Foundation (20060106) for Yong Teachers of Northeast Normal University.
文摘Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tori X, Y with A belong X, B belong Y. Let γ be a simple closed curve on F which is disjoint from A ∪ B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hγ^n（C） ： n ∈ Z} is the set containing all the simple closed curves on F which is separating with respect to （A, B）, where hγ is the Dehn twist along γ on F. This also shows how two simple closed curves on F which are separating with respect to （A, B） are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.
