Varieties via a filtration of the KBSM and knot contact homology

Abstract

This paper explains a research on the character varieties and the Kauffman bracket skein module (KBSM) of knot exteriors, which has been done by the author in his stay at the University of California, Riverside, 2004-2006. As a main consequence, for any 2-bridge knot, we gave a representation theoretical proof to the conjecture that degree 0 abelian knot contact homology HC0ab(K) of a knot K in the 3-space R3 is isomorphic to the SL2(C)-character ring of the fundamental group of the 2-fold branched cover of the 3-sphere S3 branched along K. (This result was first shown by L. Ng [23] by using cord formalism.) Based on this result, deep studies of the conjecture has been done in [18–20].