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In this essay we focus on the question of whether a 3-manifold (possibly with bound- ary) like M supports a codimension-1 transversely oriented foliation like F , such that F is transverse to ∂M and does not have Reeb components. If such foliation exists, then ∂M is necessarily a (possibly
empty) union of tori and (based on the works of Novikov, Reeb and Rosenberg) M is either S2 × S1(with F being the prod- uct foliation) or M is irreducible. In a paper by David Gabai, which we discuss, the sufficiency of these conditions is proved in case of non-trivial second homology group.Furthermore, from the works of Thurston it is concluded that compact leaves of such foliation are norm minimizing... 

In this thesis, we study a homological invariant in Knot theory, called Khovanov homology. The main property of this invariants is that it gives us the Jones polynomial, as its graded Euler characteristic. Besides, the functor (1+1) TQFT, from the category of closed one-manifolds to the category of vector spaces is employed in its construction. By making some changes to this functor and defining another functor and some other steps, the so-called Lee spectral sequence is derived which starts from Khovanov homology and converges to another homological invariant of links, called Lee-Khovanov homology. Computation of this homology is very simple. By using this spectral sequence, a numerical... 

Knot theory is the study of ambient isotopy classes of compact 1–manifolds in a 3-manifold. In classical knot theory this 3-manifold is R3 or S3. This field has experienced a great transformative advances in recent years because of its strong connections with and a number of other mathematical disciplines including topology of 3-manifolds and 4-manifolds, gauge theory, representation theory, categorification, morse theory, symplectic geometry and the theory of pseudo-holomorphic curves. In this thesis we start with classical knot theory, introducing some of its (classical) invariants like unknotting number, Seifert genus and slice genus of a knot, knot group and finally Alexander Polynomial...