The Burau representation is not faithful forn= 5

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...

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.

Entropy and the Burau Representation

The topological entropy of a braid is the infimum of the entropies of all homeomor-phisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius...

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(...