M.Sc Thesis


M.Sc StudentBen Yehezkel Tzlil
SubjectClosed Incompressible Surfaces in 3-Bridge Knot and
link Complements
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Yoav Moriyah
Full Thesis textFull thesis text - English Version


Abstract

In this thesis, we study an infinite set of 3-bridge, prime, non-split links in the 3-sphere, which are represented by plat diagrams. The set is composed of links which have 6-plat diagrams of odd length greater

than 3, the crossings in each twist box have the same parity, and the absolute value of the number of crossings, in each twist box is bigger than two. For such links, we prove that there are no closed, pairwise incompressible surfaces in their complements. The proof is obtained by studying the intersections of the surface with the projection plane of the plat.


The existence of properly embedded surfaces in a three dimensions manifold has significant implications for the manifold structure. Thus, the nature of these surfaces is of great interest. For example, Menasco proved that for a prime, non-split link which has an alternating projection there are no closed, pairwise incompressible surfaces in its complement. Moriah and Finkelstein proved that knots or links with 2m-plat projections of length greater than 5 which have more than three crossings in each twist box, contain many closed incompressible surfaces in their complements. In this thesis we generalize Menasco's theorem by replacing the alternating projection restriction with the 6-plat restrictions.