Chord Diagrams, d-diagrams, and Knots
by
Vassily Manturov
Moscow State Univ., Ind. Univ. of Moscow

Bracket Semigroup of Knots and The Algebra of Chord Diagrams.

In the present paper a new way for coding knots ¹ Each knot is encoded by means of so-called proper double-bracket structures- some words in alphabet of [ , ], (, and ).

Now describe knot semigroup K.

i) Elements of K are isotopty classes of links with fixed point on oriented component.

ii) Unity of K is the isotopy class of unknot.

iii) Consider two elements A and B (we choose representatives of these classes). Let A₁, A₂ be points on A in the small neighbourhood U of the initial poin such that A₁ is after A₂ according to the component orientation. Analogously choose B₁, B₂ at some representative B.

Consider an immersion of A and B to the plane, such that initial points of A and B are near each other and there exists a plane p such that p \cap A, p \cap B in U and A\(p \cap A) and B\(p \cap B) are located at the different sizes of p. Delete arc A₁A₂ from A and arc B₁B₂ from B and identify A₁ with B₂ and B₁ with A₂. Thus we obtained a link L with inititial point A₁=B₂ and orientation coinides with that of A and B. We choose class of L for representing AB.

Definition 1. Word A in four-bracket alphabet ( , ), [ , and ] is called proper bracket structure if words A_[   ] and A_(   ), obtained from A by deleting round (corr. square) brackets are proper bracket structure in a proper sence.

This coding allows to give a simple algebraic description of the knot semigroup. This group is a subgroup of finitely-generated tangle semigroup-semigroup generated by (,    ),    , [,    ] with some relations. Elements of semigroup G are equivalence classes² of proper bracket structures in this alphabet.

The main result of the work consists in construction of a canonic isomorphism between algebraic semigroup G and geometrically generated semigroup K.

Definition 2. Chord diagram ³, is called d-diagram if its chord can be divided into two sets of non-intersecting chords.

The construction of canonic isomorphism is closely connected with d-diagrams , i.e. first we construct an isomorphism of K and a d-diagrams semigroups and then apply it for G.

This way for encoding classical knots by means of d-diagrams can be appliable for the case of all chord diarams and so-called örientable" virtual knots.

Some modernization of the notion of d-diagrams allows to encode all singular knots up to isotopy. That allows to encode the algebra of chord diagrams factorized by 1T and 4T-relations in terms of chord diagrams, or words in some alphabet.

Namely, consider the following algebra DD.

i) Elements of this algebra are linear combinations of C-equivalency of knot (not link) d-diagrams with hard chords⁴.

ii) Unity is chord C-equivalency class of d-diagram without hard chords.

iii) Multiplication is defined on d-diagrams as follows:

Besides C-equivalency, there exist the following relations:

iv) ("1T-relation").

Picture Omitted
v) ("4T-relation").

Picture Omitted
Denote the algebra of chord diagrams, factorized by 1T and 4T-relations by A^c

Main theorem 2. Algebra DD is isomorphic to A^c.

References:

[Ma1] V.O.Manturov. Bifurcations, Atoms, and Knots (in Russian)// Vestnik of the MSU, ser. math., 1999, to be appeared

[Ma2] V.O.Manturov. Atoms, Vertical Atoms, Chord diagrams, and Knots. Enumeration of Atoms of Low Complexity Using Mathematica 3.0(in Russian)// Topological Methods in Hamiltonian Systems Theory, Factorial ed., Moscow., 1998 pp. 203-212

[Ma3] V.O.Manturov. Bracket Semigroup of Knots.(in Russian)// Math. Zametki., 1999, to be appeared

Footnotes:

¹Under "knot" we mean a link with fixed point on an oriented component, until otherwise is set; for example, oriented knot with a fixed point on it.

²There exists 8 equivalence relations; they are ommitted

³formal, non-related to singular knots

⁴We use bald lines for hard chords and dotted lines for other chords

Date received: June 2, 1999