Characteristic varieties of algebraic curves

Invariants of the fundamental group of complements to plain curve complex singularities called Characteristic Varieties are considered. Characteristic varieties and corresponding ideals and polytopes of quasiadjunction are the finer invariant of algebraic knots and links than the Alexander polynomial. Applications of characteristic varieties and polytopes of quasiadjunction to computations of log-canonical threshold of divisors are considered. A formula for computing the log-canonical threshold of plane curve singularities with transversal branches is derived using ideals of quasiadjunction. In addition, the computational procedure for finding the polytopes of quasiadjunction is considered.