Some Questions in Low-Dimensional and Contact Topology

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This project aims to carry out research on several interrelated topics in contact and low-dimensional topology. A "contact structure'' is an extra geometric structure on a manifold (these structures originate from physics); the study of contact structures and knots in contact manifolds draws on ideas from different fields (geometry, algebra, combinatorics). This project will contribute to the development of these areas, especially to geometry of manifolds. The main part of the project concerns the investigation of "right-handed'' and "left-handed'' properties of certain knots. Related ideas often appear in the broader context of geometry, algebra, and physics.This project will focus on two different topics. The first topic concerns knot theory and contact topology in dimension 3, and is related to knot invariants of gauge-theoretic and combinatorial nature. The project's goal is to develop properties and applications for a special class of transverse knots (those with certain "positivity" properties); the PI expects to find relations between algebraic properties of corresponding braids, behavior of invariants from knot Floer and Khovanov homology (including an invariant introduced by the PI in 2006), properties of symplectic surfaces (cobordisms) connecting the knots, and rigidity of their branched double covers. This investigation develops a novel approach (with a focus on braid group orderings and the braid monodromy) but also continues the PI's past research. The PI also plans to continue studying flexible phenomena in higher-dimensional contact topology and expects to contribute to this rapidly developing area. The goal is to better understand properties of rigid/flexible contact manifolds, developing analogs of existing results in dimension 3.