Knotted surfaces, Homological Norm and Extendable Subgroup

TL;DR

This paper constructs a specific knotted surface in four-dimensional space with a finitely generated group of symmetries acting on its first homology, using a new homological norm to analyze its properties.

Contribution

It introduces a novel homological norm on the first homology of knotted surfaces in S4 and demonstrates its additivity, leading to the construction of surfaces with finitely extendable symmetries.

Findings

Existence of knotted surfaces with finitely many extendable symmetries

Definition of a new homological norm with additive properties

Construction applicable to surfaces of arbitrary genus

Abstract

We prove that for arbitrary g, there is a surface K of genus g embedded in S4, which has finitely many extendable self-homeomorphisms' action on H1(K,Z), by defining a norm on H1(K,Z) and proving its additivity.