New Permutation Representations of the Braid Group
  We give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S_{mk} for all k and m \geq 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\leq l < m, we prove in particular that if \frac{m}{l} is odd then there are 1 + \frac{m}{l} non-conjugate homomorphisms included in our family.   We define a certain natural restriction on homomorphisms B_k to S_n, common to all homomorphisms in our family, which we term 'good', and of which there are two types.   We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in the infinite family of homomorphisms we gave. For m=3, we prove that all good homomorphisms B_k to S_{3k} of type 2 are also included in this family.   Finally, we refute a conjecture made by Matei and Suciu regarding permutation representations of braids and give an updated conjecture. 
Amiel Ferman, Tahl Nowik, Robert Schwartz, Mina Teicher