A Note on the Kernel of the Gassner Representation

Abstract

This note is an appendix of Goto’s Master Dissertation³⁾. The subject is THEOREM 3.14 of Birman’s book¹⁾ which states that the kernel of the Gassner representation is a subset of the commutator subgroup of the pure braid group. When we prepared Goto’s Master Dissertation, we could not fill the space between the lines in the proof. THEOREM 3.14 deals with not only the kernel of Gassner representation but also the kernel of Burau representation which is a homomorphism from the braid group. The proof for Burau representation is done by considering the determinants of the image of the representation. The determinants are Laurant polynomials of one variable. It is stated in the proof that the case of Gassner representation is proved similarly. Since the determinants of this case are Laurant polynomials of multi variables, we need more consideration. After Goto’s Master Dissertation, we complete the proof for Gassner representation by elementary matrix calculation as we show here.