Many Three Dimensional Objects Inspired From Finite Groups

TL;DR

This paper introduces new algebraic structures derived from finite groups that produce link and 3-manifold invariants, extending existing frameworks with generalized R-matrices and colored link invariants.

Contribution

It develops extended R-matrices from finite groups, introduces new invariants for links and 3-manifolds, and constructs groups that dominate these invariants, proving their invariance.

Findings

New R-matrices derived from finite groups produce braid group actions.

Defined extended R-matrices that generalize Turaev's enhanced R-matrix.

Constructed groups that serve as invariants for links and 3-manifolds.

Abstract

Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group $G$, we find some quite simple $R-$matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "extended $R-$matrix" and "generalized extended $R-$matrix" generalizing Turaev's enhanced $R-$matrix, which can still give invariants of oriented links. With these new frames, we show that above $R-$matrix, together with certain commuting pairs (essentially conjugacy classes of commuting pairs ) of $G$ can give integer invariants of oriented links. We construct some group dominating these integer invariants and prove these groups are link invariant by themselves. Using the language of the (colored) tangle category, we extended above invariant to invariant of links and ribbon links colored by commuting pairs. We show given a oriented link diagram $L$, a suitable weighted sum of above invariant on all kinds of coloring of $L$ (by conjugacy classes of $G$) is invariant under both two types of Kirby moves, thus giving a invariant for closed three manifolds. We define a group dominating those invariants, and prove this group is a three manifold invariant by itself.