In this talk, we will introduce an algebraic variety
in an affine space over the complex number field
constructed via the Kauffman bracket skein module (KBSM)
of a knot exterior. One of the main ideas for the
construction of the variety is that the polynomial map
from the affine space to itself can be defined
by using a representation of the braid group into
the endomorphisms of the KBSM of a handlebody.
In fact, the algebraic variety turns out to be
an invariant of knots in 3-sphere.
In this talk, we will try to get a better understanding
of the variety by focusing on the number of its
irreducible components. Then we will see a relationship
of the variety with so-called the Casson-Lin invariant
defined by X-.S-. Lin, which in fact inspired the above
main idea, and moreover a relationship of the variety
with the highest degree of the A-polynomial in terms of $L$.