The faithfulness question for the Burau representation

Abstract

We consider the Burau and Gassner representations of the braid groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{B_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A sufficient condition for faithfulness, involving just a pair of arcs, is shown to be necessary as well for all but at most two values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the Burau case, this implies nonfaithfulness for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-slanted-equals 10"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geqslant 10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .