An expository report on the paper ''From Goeritz matrices to quasi alternating links" by Józef H. Przyticki

Abstract

In this project, we aim to study a pathway that starts from the Tait Graph and the Goeritz matrix of a given link to define what is the family of quasi-alternating links and study some of its important properties. For a given link diagram, we defined a Goeritz matrix whose determinant is an invariant of knots up to ambient isotopy. Also, the signature and nullity of Goeritz matrix have a direct relation to the signature and nullity of the link (both of which are invariants of links). The link diagram helps us construct the Seifert surface, and we subsequently calculate the Seifert matrix for the surface, which provides us with the tools of calculating the Alexander polynomial, and the Conway polynomial. We also calculated the Tristam-Levine signature, which is a link invariant. Then we defined Quasi-alternating links and then studied some of the properties of the quasi-alternating links. We further moved to defining the QACTI which is a measure of the depth or complexity of quasi-alternating links. With the help of the tools we studied and developed in the thesis, we then found some upper bounds and lower bounds for the QACTI of a quasi-alternating link to get an approximation of QACTI of the quasi-alternating link.