395 research outputs found
Cabling Burau Representation
The Burau representation enables to define many other representations of the
braid group 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 . In particular, we show that
there is no gain in terms of faithfulness by cabling the Burau representation.Comment: 11 page
On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants
We study the Chern-Simons topological quantum field theory with an
inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple
one by taking its semi-direct product with its Lie algebra. We find that the
standard knot observables (i.e. traces of holonomies along knots) essentially
vanish, but yet, the non-semi-simplicity of our gauge group allows us to
consider a class of un-orthodox observables which breaks gauge invariance at
one point and which lead to a non-trivial theory on long knots in
. We have two main morals : 1. In the non-semi-simple case, there
is more to observe in Chern-Simons theory! There might be other interesting non
semi-simple gauge groups to study in this context beyond our example. 2. In our
case of an inhomogeneous gauge group, we find that Chern-Simons theory with the
un-orthodox observable is actually the same as 3D BF theory with the
Cattaneo-Cotta-Ramusino-Martellini knot observable. This leads to a
simplification of their results and enables us to generalize and solve a
problem they posed regarding the relation between BF theory and the
Alexander-Conway polynomial. Our result is that the most general knot invariant
coming from pure BF topological quantum field theory is in the algebra
generated by the coefficients of the Alexander-Conway polynomial.Comment: To appear in Journal of Mathematical Physics vol.46 issue 12.
Available on http://link.aip.org/link/jmapaq/v46/i1
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