The Density of Lawrence-krammer and Non-conjugate Braid Representations of Links
نویسنده
چکیده
We use some Lie group theory and Budney’s unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.
منابع مشابه
Twisted Alexander Polynomial for the Lawrence-krammer Representation
In this paper, we prove that the twisted Alexander polynomial for the Lawrence-Krammer representation of the braid group B4 is trivial. This gives an answer for the problem whether the twisted Alexander polynomial for given faithful representations is always non-trivial.
متن کاملA Family of Representations of Braid Groups on Surfaces
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
متن کاملIrreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$. We specialize the indeterminates used in defining these representations to non zero complex numbers. We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$. We then determine necessary and sufficient conditions that guarantee the irreducibility of th...
متن کاملThe Lawrence–krammer Representation Is Unitary
We show that the Lawrence–Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence–Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawr...
متن کاملInvariant Subspaces of the Lawrence–krammer Representation
The Lawrence–Krammer representation was used in 2000 to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence–Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values. It is also known that when the representation is reducible, t...
متن کامل