A note on affine links

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.