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Title: Relative Symplectic Caps, Fibered Knots And 4-Genus
Authors: Kulkarni, Dheeraj
Advisors: Gadgil, Siddhartha
Keywords: Symplectic Geometry
Symplectic Capping Theorem
Symlpectic Manifolds
Fibered Knots
4-Genus Knots
Symplectic Caps
Knot Theory
Contact Geometry
Contact Manifolds
Quasipositive Knots
Symplectic Convexity
Symplectic Neighborhood Theorem
Seifert Surfaces
Riemann Surface
Submitted Date: Jul-2012
Series/Report no.: G25244
Abstract: The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus. In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S3 . We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in S3 . Using this result, we give a criterion for quasipostive fibered knots to be strongly quasipositive. Symplectic convexity disc bundles is a useful tool in constructing symplectic fillings of contact manifolds. We show the symplectic convexity of the unit disc bundle in a Hermitian holomorphic line bundle over a Riemann surface.
Abstract file URL: http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf
URI: http://etd.iisc.ernet.in/handle/2005/2285
Appears in Collections:Mathematics (math)

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