A note on affine links

TL;DR

This paper explores the relationships between affine and classical braid groups, providing new algebraic insights and a novel definition of affine links through injections and surjections, with implications for link theory.

Contribution

It establishes a surjection from the affine braid group to the classical braid group, detects its kernel, and introduces an injection of the B-type into the A-type braid group for defining affine links.

Findings

The affine braid group surjects onto the classical braid group.

The kernel of the surjection is characterized using Schreier's Theorem.

Affine links can be defined via closures of affine braids with Markov conditions.

Abstract

We view the $\tilde{A}$-type affine braid group as a subgroup of the $B$-type braid group. We show that the $\tilde{A}$-type affine braid group surjects onto the $A$-type braid group and we detect the kernel of this surjection using Schreier's Theorem. We then describe an injection of the $B$-type braid group into the $A$-type braid group which allows us finally to give a definition of affine links, as closures of affine braids viewed as A-type braids after composing the above injections, and we prove that the two conditions of Markov are necessary and sufficient to get the same affine closure of any two affine braids.