Study of positive knots via contact structures

2017 Fiscal Year Final Research Report

• Project/Area Number: 16H07230
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Tagami Keiji 東京理科大学, 理工学部数学科, 助教 (60778174)
• Project Period (FY): 2016-08-26 – 2018-03-31
• Keywords: 結び目 / 正結び目 / 接触構造 / ラグランジアン充填

Outline of Final Research Achievements

A knot is a smooth embedding of a circle into the 3-dimensional Euclidean space. A knot is Lagrangian fillable if it bounds an oriented Lagrangian surface from below in the symplectisation of the standard contact structure of the 3-dimensional Euclidean space.
Hayden and Sabloff proved that any positive knot is Lagrangian fillable. Inspired by their work, in this study, I investigated relations between the Lagrangian fillability and the positivity of knots. As a result, I proved that (1) any almost positive knot with a certain condition on its Seifert graph is Lagrangian fillable, (2) any alternating and Lagrangian fillable knot is positive.

Free Research Field

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