On Knotted Subgroups

TL;DR

This paper introduces the concept of knotted subgroups within Lie groups, characterizes them through infinitesimal elements, and provides classifications for specific groups like SU(2), SU(3), SL(2,R), and SL(3,R).

Contribution

It defines knotted subgroups in Lie groups, characterizes them via infinitesimal elements, and offers complete classifications for certain groups using spectral and Jordan canonical forms.

Findings

Characterization of knotted subgroups in terms of infinitesimal elements.

Complete classification of knotted subgroups in SL(2,R) and SL(3,R).

Examples provided for SU(2) and SU(3).

Abstract

In this article, we defined a knotted subgroup of a Lie group and considered a geometric notion of equivalence among them. We characterized these knotted subgroups in terms of one-parameter subgroups and provided examples in the case of SU(2) and SU(3). Infinitesimal elements that give rise to knotted subgroups of SU(n) and SO(n) are characterized as well. Canonical forms for their knotted subgroups are presented and their properties are described in terms of the spectrum of the corresponding infinitesimal elements. Finally, knotted subgroups of SL(2,R) are completely classified using direct computation while knotted subgroups of SL(3,R) are completely classified using Jordan canonical forms.