Variations of Right-Veering Open Books and Knot Positivity

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The project will advance interaction of the theory of knots and links (knots with several components) and contact topology, two significant subjects in Geometry and Topology. The project will progress the study of braids (links that look like actual braids) that has numerous applications to Algebraic Geometry, Operator Algebras, Homotopy Theory, Robotics, Cryptography, Geometry and Topology. The fundamental issue the project seeks to address is using diagrams on the plane way to detect whether contact structures are tight or overtwisted. Some parts of the project will involve research by graduate students. The Kids Topology Club will engage young Iowans who have potential to contribute to the STEM research of the US. Project goals are: (1) to develop a diagrammatic method to detect tightness or overtwistedness of a given contact 3-manifold, (2) to extend the idea of (1) to knot theory (detection of non-looseness or looseness of a transverse link), (3) to compare various positivities of knots and links from braid theory and contact geometry view points, and find their geometric meanings. Main methods used are the Bennequin-Eliashberg inequality, the Giroux correspondence, Birman and Menasco’s braid foliations, and Ito and Kawamuro’s open book foliations. To approach the goals, properties of twist-left-veering mapping classes and the defects of the 3- and 4-dimensional Bennequin-Eliashberg inequalities will be investigated. Scope of the project is (1) to study relation between twist-left-veering that Ito-Kawamuro introduce and inconsistency that Wand defined, (2) to continue investigating the depth of transverse knots via twist-left-veering, and (3) to study relation between the above-mentioned defects and Heegaard-Floer and Khovanov invariants. Potential contribution is to advance the study of (1) tight contact structures that is a central subject in contact geometry and has strong connection to symplectic geometry and algebraic geometry, and (2) quasipositive knots that are important in topology and algebraic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.