Braid representations

The Lawrence-Krammer representation is a quantization of the symmetric square of the Burau representation

The Lawrence–Krammer representation gives a faithful linear realisation of every braid group. We demonstrate that by replacing natural numbers with q-numbers and using the q-analogue of Pascal’s triangle, which naturally appears in braid group representations, the Lawrence-Krammer representation arises as a ‘quantisation’ of the symmetric square of the Burau representation and yields new braid group representations.