Representations of small braid groups in Temperley–Lieb algebra

Abstract

In this paper we consider the question of faithfulness of the Jones' representation of braid group B n into the Temperley–Lieb algebra TL n . The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493–505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B 6 into TL 9 , 2 —to the authors' knowledge the first such examples in the second gradation of the Temperley–Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.