Petal diagrams of knots are projections of knots to the plane such that the diagram has exactly one crossing. Petal diagrams offer a convenient and combinatorial way of representing knots via their associated petal permutation. In this thesis we study the fundamental group and Seifert surfaces of knots in petal form. Using the Seifert-Van Kampen theorem, we give a group presentation of the fundamental group of the knot complement of a knot in petal form. We then discuss Seifert surfaces and use decomposition diagrams to represent the Seifert surfaces of knots in petal form. We finally give an algorithm to produce a set of decomposition diagrams for a spanning surface of a knot in petal form and prove that for incompressible surfaces such decomposition diagrams are unique up to perturbation moves.

We study Seifert surfaces for links, and in particular the Kakimizu complex MS(L) of a link L, which is a simplicial complex that records the structure of the set of taut Seifert surfaces for L. First we study a connection between the reduced Alexander polynomial of a link and the uniqueness of taut Seifert surfaces. Specifically, we reprove and extend a particular case of a result of Juhasz, using very different methods, showing that if a non-split homogeneous link has a reduced Alexander polynomial whose constant term has modulus at most 3 then the link has a unique incompressible Seifert surface. More generally we see that this constant term controls the structure of any non-split homogeneous link. Next we give a complete proof of results stated by Hirasawa and Sakuma, describing explicitly the Kakimizu complex of any non-split, prime, special alternating link. We then calculate the form of the Kakimizu complex of a connected sum of two non-fibred links in terms of the Kakimizu complex of each of the two links. This has previously been done by Kakimizu when one of the two links is fibred. Finally, we address the question of when the Kakimizu complex is locally infinite. We show that if all the taut Seifert surfaces are connected then MS(L) can only be locally infinite when L is a satellite of a torus knot, a cable knot or a connected sum. Additionally we give examples of knots that exhibit this behaviour. We finish by showing that this picture is not complete when disconnected taut Seifert surfaces exist.

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.