Manchester Geometry Seminar 2002/2003

A 3-page presentation method was introduced in knot theory by Dynnikov in 1998. This method uses embeddings of links into a book with 3 pages. Such embeddings are encoded by words in a 12-letters alphabet. Also there are only finitely many local moves on these words generated by any ambient isotopy of links in the three-space. More exactly, an algebraic semigroup DS with 12 generators and 48 relations was constructed, such that there is a one-to-one correspondence between all isotopy classes of non-oriented links in the three-space and all central elements of the semigroup DS [1]. The generators and relations of DS are quite simple and have clear geometrical meaning. So, the 3-page presentation method has already supplied a purely algebraic interpretation of the pure topological problem of the classification of links in the three-space up to isotopy. In 2001 the speaker generalized Dynnikov's approach to knotted 3-valent graphs. For such graphs the universal semigroup has 24 generators and 90 relations [3]. In 2002 Vershinin and the speaker also extended the 3-page presentation to singular knots. For singular knots the universal semigroup has 15 generators and 84 relations [4]. Recently Dynnikov announced an algorithm for recognizing split and composite links, and the unknot, exploiting a book presentation [2].

1. I. A.Dynnikov, Finitely presented groups and semigroups in knot theory. Proc. Steklov Inst. Math. 2000, no. 4 (231), 220-237.
2. I. A.Dynnikov, Arc presentation of links. Monotonic simplification (preprint), available at http://front.math.ucdavis.edu/math.GT/0208153
3. V. A. Kurlin, Three-page Dynnikov diagrams of linked 3-valent graphs. Functional Analysis and Its Applications, 35 (2001), no. 3.
4. V. A. Kurlin, V. V. Vershinin, Three-page embeddings of singular knots, to appear in Functional Analysis and Its Applications.