Torus Knots and Contact Surgeries

Abstract

A closed curve homeomorphic to the unit circle in a 3-manifold is called a knot. In particularly, a knot that can be drawn on a torus without intersecting itself is called a torus knot. In this thesis, we study torus knots and their topological properties, invariants and polynomials. We study Dehn surgery on torus knots in topological 3 manifolds. Then, we study contact 3-manifolds. We study a special class of Legendrian knots which have topological knot type as torus knots. The aim of this thesis is to study lens spaces by using contact surgery techniques. For this purpose, obtaining lens spaces L(4m+3,4) by Legendrian surgery along the negative torus knots T(2,−(2m+1)) where m ≥ 1 are studied in detail.