2002/02/21 by Won Taek Song, Song, Won Taek
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Geometric and Algebraic Topology #math.GT #math.RT #msc:20F36 #msc:57M07
paper · pdf · doi:10.48550/arxiv.math/0202214
16 pages, 53 pictures
arxiv created 2002/02/21 · arxiv updated 2009/11/30
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 Lawrence--Krammer matrix is invariant under substitution of its variables with their inverses up to multiplication by units, and is not a complete conjugacy invariant for braids.