A Braid Representation and Corresponding Closed Braid Invariant

Abstract

In [5] Xiao Song Lin shows that “traceless” representations of a braid group into SU(2) counted with appropriate sign gives one half of the signature of the associated knot. This is done with a skein technique, comparing the behavior of the representations and the sign of the Conway polynomial of the knot evaluated at 2i. These results appear somewhat mysterious. In this paper, we present a representation of the braid group into SL(2n,Z) where n+1 is the number of strands of the braid. This matrix arises naturally from considering the two-fold branched cover of a ball containing the braid. The lower right n×n block is basically the reduced Brauer representation evaluated at –1 while the lower left n×n block of this matrix has special significance. The absolute value of the determinant of this block corresponds to the determinant of the corresponding closed braid. If one defines C(η) to be the determinant of this block normalized properly one gets an invariant of the corresponding closed braid which is equal to the Conway polynomial of the corresponding closed braid evaluated at 2i. Therefore, one can use a skein technique to compute the signature of the knot from C(η). Since the absolute value of the Conway polynomial evaluated at 2i is the order of the homology of the two-fold branched cover of a link, C(η) also contains this homological information. It will be shown in [6] that the lower left n×n block is a presentation matrix of the first homology of the two-fold cover. This homology matrix is related to the traceless representations of Xiao Song Lin that come from the homology of the two-fold branched cover.