Skein module deformations of elementary moves on links

Abstract

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 (see [12,26,36]).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.

Bottom tangles and universal invariants

“…Twist moves have long been studied in knot theory. For a recent survey, see Przytycki [70]. Here we give a few examples from the literature with translations into our setting.…”

Three geometric applications of quandle homology

“…Of particular interest in knot theory are 4-moves. One of the reasons is the following old conjecture [Kir,Pr1,Pr2].…”

“…Not every link is 4-move equivalent to a trivial link, in particular, the linking matrix modulo 2 is preserved by 4-moves. Furthermore, Nakanishi demonstrated that the Borromean rings cannot be reduced to the trivial link of three components [Nak,Pr2]. Kawauchi expressed the question for links as follows:…”

“…Proof. It is known (see for example [Pr1,Pr2,NP1]) that the dihedral quandle R n is an invariant under n-moves. In other words, for any R n -coloring of the two strings with n half-twists, the colors of the two initial arcs are the same as colors of the corresponding final arcs (see Figure 9 for an illustration of this fact in the case of 4-move).…”

A Lagrangian representation of tangles II

“…For string links, it is equivalent to a construction of Le Dimet [5]. We refer to [6,7] and references therein for related work on invariants of tangles.…”