  This paper is based on my talks (`Skein modules with a cubic skein relation:
properties and speculations' and `Symplectic structure on colorings, Lagrangian
tangles and its applications') given in Kyoto (RIMS), September 11 and
September 18 respectively, 2001. The first three sections closely follow the
talks: starting from elementary moves on links and ending on applications to
unknotting number motivated by a skein module deformation of a 3-move. The
theory of skein modules is outlined in the problem section of these
proceedings.
  In the first section we make the point that despite its long history, knot
theory has many elementary problems that are still open. We discuss several of
them starting from the Nakanishi's 4-move conjecture. In the second section we
introduce the idea of Lagrangian tangles and we show how to apply them to
elementary moves and to rotors. In the third section we apply (2,2)-moves and a
skein module deformation of a 3-move to approximate unknotting numbers of
knots. In the fourth section we introduce the Burnside groups of links and use
these invariants to resolve several problems stated in section 1.
