The Burau representation is not faithful for n ≥ 6

The Burau Representation is Not Faithful for n = 5 Stephen

The Burau representation is not faithful for n = 5

The Burau representation is a natural action of the braid group Bn on the free Z[t; t−1]{module of rank n−1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n = 3. Moody has shown that it is not faithful for n 9 and Long and Paton improved on Moody’s techniques to bring this down to n 6. Their construction uses...

THE BURAU REPRESENTATION OF THE BRAID GROUP Bn IS UNFAITHFUL FOR LARGE n

Two fundamental theorems of classical knot theory, by J. Alexander, are that every knot is a closed braid, and second, that a certain procedure (see [1,2] for both results) assigns to each knot K a polynomial A(K) in T, which only depends on the topological type of the knot once normalized by a power of T to take a positive value at T = 0. With this normalization, take T to be a transcendental ...

The Faithfulness Question for the Burau Representation

We consider the Burau and Gassner representations of the braid groups B„ . 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 n. In the Burau case, this implies nonfaithfulness for n > 10 . Alexander showed in [6] that any knot K can be made by connecting the ends of some braid b. The Alexander polynomial A(...

Cabling Burau Representation

The Burau representation enables to define many other representations of the braid group Bn by the topological operation of “cabling braids”. We show here that these representations split into copies of the Burau representation itself and of a representation of Bn/(Pn, Pn). In particular, we show that there is no gain in terms of faithfulness by cabling the Burau representation.