New Permutation Representations of The Braid Group

We give a new infinite family of group homomorphisms from the braid group Bk to the symmetric group Smk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms Bk → Sn, common to all homomorphisms in our family, which we term good, and of which there are two types. We prove that all good homomorphisms Bk → Smk of type 1 are included in the infinite family of homomorphisms we gave. For m = 3, we prove that all good homomorphisms Bk → S3k of type 2 are also included in this family. Finally, we refute a conjecture made in [MaSu05] regarding permutation representations of braids and give an updated conjecture.