Abstract
Quasipositive surfaces originally arose in the study of complex plane curves in the '80s. They were originally defined by Lee Rudolph in a topological manner, as the Bennequin surface of a strongly quasipositive link. Lee Rudolph and others have studied these surfaces using tools from algebraic topology. In more recent years many connections have been made between quasipositive surfaces and contact geometry including a definition of quasipositive surfaces as the ribbon surface of a Legendrian graph. Beyond this classification many things are known about these surfaces, for instance they satisfy the Bennequin and Slice-Bennequin inequalities, yet many things remain a mystery. Outside of fibered knots it is not known whether the strong quasipositivity of a link guarantees the quasipositivity of a minimal genus Seifert surface. In the case of fibered knots where there is a unique minimal genus Seifert surface, the strong quasipositivity of a link guarantees that this minimal gensu Seifert surface is quasipositive. Outside of this case strongly quasipositive links can have distinct non-isotopic minimal genus Seifert surfaces. Baker and Motegi raised the question of whether strongly quasipositive links can have minimal genus Seifert surfaces which are not quasipositive. This question can be refined with the contact characterization of quasipositive surfaces. If a link bounds a surface which is the ribbon of a Legendrian graph must every minimal genus Seifert surface be isotopic to the ribbon of a Legendrian graph. As stated this question doesn't take advantage of the contact structure. The boundary of a Legendrian ribbon can be given a natural structure as a transverse link, to use the contact structure we can ask the question: if a link bounds a surface which is the ribbon of a Legendrian graph must every minimal genus Seifert surface be isotopic to the ribbon of a Legendrian graph via an isotopy preserving the transverse link type of its boundary? The answer to this question is no. In this dissertation we construct a transverse link in the universally tight contact structure on L(4,1) which bounds a Legendrian ribbon and another surface which cannot be isotoped into a Legendrian ribbon preserving the transverse link type of its boundary. In order to prove the non-existence of this isotopy we use convex surface theory. In particular we analyze the bypasses near the surface to find that there can be no such transverse isotopy.