In this thesis, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus of a Legendrian knot L in a contact 3-manifold as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framings given by the contact structure and the page agree. For any topological link in 3-sphere we construct a planar open book decomposition whose monodromy is a product of positive Dehn twists such that the planar open book contains the link on its page. Using this, we show any topological link, in particular any knot in any 3-manifold M sits on a page of a planar open book decomposition of M and we show any null-homologous loose Legendrian knot in an overtwisted contact structure has support genus zero.

In this thesis, we write explicitly the open book decompositions of links of quotient surface singularities that support the corresponding unique Milnor fillable contact structures. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the corresponding unique Milnor fillable contact structures. That minimal page-genus is called Milnor genus. In this thesis we also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus of the corresponding contact structure. For the remaining we are able to find an upper bound for the support genus which would be a step forward in understanding these contact structures.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21545号 / 理博第4452号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 大槻 知忠, 教授 向井 茂, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM