Construction and manipulation of Seifert surfaces in knot theory (a note in 2023) (Intelligence of Low-dimensional Topology)

Abstract

In this note, I present a survey of construction and manipulation of Seifert surfaces in knot theory, including, fiber surfaces with hidden Hopf plumbings (with Murasugi), fiber surfaces with arbitrary enhanced Milnor number (with Rudolph), new fiber-producing twisting operation (with Van Quach), Seifert surfaces preserved by a strong involution (with Sakuma and Hiura) and so on. As an application, given is a demonstration of finding a strongly invertible diagram by manipulating a fiber surface. We start with updating the Seifert algorithm by omitting the step of applying arrows to segments in a knot projection.