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.
