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 $L$.