ISSN 1063-7850, Technical Physics Letters, 2006, Vol. 32, No. 5, pp. 445–448. © Pleiades Publishing, Inc., 2006.Original Russian Text © S.A. Grishanov, V.R. Meshkov, A.V. Omel’chenko, 2006, published in Pis’ma v Zhurnal Tekhnichesko

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Fiziki, 2006, Vol. 32, No. 10, pp. 61–67.

445

For a long time, the properties of spatial configura-tions with knots and entanglements received only theinterest of mathematicians. Knot theory, being to a con-siderable extent of “physical” origin, has grown into anindependent direction in topology. However, recentdecades have revealed the deep relations of this theorywith various fields in physics. In particular, it wasestablished that the Young–Baxter equation, playing animportant role in statistical physics and quantum the-ory, also appears in knot theory, where it is related toone of the so-called Reidemeister moves—transformswhich do not change the topological types of knots. Theestablishment of this relationship led to the discoveryof new powerful invariants in knot theory, such as poly-nomials of the Jones, Kauffman, and HOMFLY types[1–3]. At the same time, the physics received new ana-lytical methods.

Doubly periodic braided structures (referred tobelow as 2-braids), as well as knots and entanglements,are frequently encountered in applications. In particu-lar, two-dimensional models in statistical physics areconsidered on such lattices [4] and numerous examplesof “real” 2-braids are offered by textiles. Investigationinto the topological properties of 2-braids is of consid-erable interest for such applications. Despite this signif-icance, 2-braids have not been specially considered astopological objects so far and, in particular, no classifi-cation of 2-braids has been developed.

The main tool for the investigation of knots andentanglements are invariants. The invariant is a func-tion defined on a set of knots, which has the same val-ues for equivalent knots. Two knots are consideredtopologically equivalent (isotopic) provided that one ofthem can be continuously transformed into anotherwithout self-intersections (such deformation is calledisotopy). In the case of 2-braids, it is natural to consider

the isotopic deformations that retain periodicity of thestructure.

A 2-braid (and any invariant of this structure) isfully determined once the minimum repeat element—the unit cell—is defined. Such an element can be cho-sen in an infinite number of ways; an isotopic invariantof the given structure must be independent of the selec-tion of a unit cell. This Letter presents a new polyno-mial invariant for 2-braids, which obeys this condition.

Knot diagrams.

Knot invariants are convenientlydetermined using diagrams representing nondegenerateplanar projections with indications of the types ofcrossings [5]. A 2-braid is naturally brought into corre-spondence with a diagram on a torus. Such a diagram isobtained by identifying the opposite sides of a unit cellchosen on a flat diagram of the given structure (Fig. 1).Apparently, this correspondence is not unique, as isillustrated in Fig. 2, since there is a one-to-one corre-spondence between the manifold of reduced fractions

±

p

/

q

and the manifold of unit cells (defined to withina shift).

The relation of equivalence on a manifold of torusdiagrams corresponding to the given 2-braid can be setin terms of torus torsions, which are defined as follows.Let us cut the torus along an arbitrary meridian, rotateone edge of the cut through 360

°

in its plane, and con-

A Polynomial Invariantof Doubly Periodic Braided Structures

S. A. Grishanov, V. R. Meshkov, and A. V. Omel’chenko

De Monfort University, Leicester, United KingdomSt. Petersburg State University of Technology and Design, St. Petersburg, 191065 Russia

St. Petersburg State Technical University, St. Petersburg, 195251 Russiae-mail: vmu@peterlink.ru

Received November 28, 2005

Abstract

—A new isotopic Kauffman-type polynomial invariant of two variables for doubly periodic braidedstructures is constructed.

PACS numbers: 04.20.Jz

DOI:

10.1134/S1063785006050221

Fig. 1.

Diagrams of knots, entanglements, and 2-braids.

446

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No. 5

2006

GRISHANOV et al.

nect the edges. This procedure will be referred to as themeridian torsion. By the same token, we can define thetorsion along a parallel. Obviously, using such meridianand parallel torsions, it is possible to transform an arbi-trary unit cell into any other unit cell.

Equivalent knots.

Two knots (or entanglements)are called equivalent if there is a family of diffeomor-phic maps

f

t

:

3

3

smoothly dependent on

t

∈

[0, 1], in which

f

0

is the identical diffeomorphism and

f

1

maps

K

1

into

K

2

(so-called isotopy relating

K

0

and

K

1

).It should be noted that the requirement of the one-to-one correspondence of

f

t

maps prohibits the self-inter-section of a knot in the course of isotopy. The conceptof a planar isotopic mapping for the knot diagrams isformulated similarly.

In terms of the knot diagrams, two unoriented knots

K

1

and

K

2

are called equivalent if and only if the

K

1

dia-gram can be transformed into the

K

2

diagram using pla-nar isotopies and the Reidemeister moves

Ω

1

,

Ω

2

, and

Ω

3

(Reidemeister theorem). For torus diagrams, theReidemeister moves

Ω

1

,

Ω

2

, and

Ω

3

can be defined inthe same manner as for the planar diagrams, and theisotopy on the torus surface can be introduced by anal-ogy to the planar isotopy.

In contrast to the case of usual knot diagrams, theequivalence of 2-braids represented by two torus dia-grams cannot be unambiguously judged using only theReidemeister moves and isotopies. Indeed, these dia-grams may correspond to two different unit cells of thesame structure. In order to eliminate the problemrelated to the ambiguity of the choice of a unit cell, thepossible transformations of torus diagrams must besupplemented by the torsions defined above. Then, thefollowing generalization of the Reidemeister theoremis valid.

Theorem.

Two torus diagrams correspond to isoto-pic 2-braids if and only if these diagrams can be trans-formed into one another using isotopies on the surface

of torus and a finite number of the Reidemeister movesand torus torsions.

Thus, any function defined on a manifold of torusdiagrams, which is not changed by torsions and Reide-meister moves, determines an isotopic invariant of2-braids. Below, we construct such a function with val-ues in the manifold of polynomials.

Isotopic invariant.

Following Kauffman [2], let usbring every torus diagram

l

into correspondence with apolynomial

⟨

l

⟩

of variables

a

,

b

, and

c

, which obeys thefollowing relations:

(1)

(2)

(3)

where

l

= ;

l

A

= ;

l

B

=

are three diagrams that are identical outside the dashedcircle and

is the circle diagram. Polynomial

⟨

l

⟩

iscalled the Kauffman bracket [5].

Let

n

be the number of crossings in diagram

l

. Byconsequently applying relation (1) to each crossing, itis possible to express

⟨

l

⟩

in terms of polynomials

⟨

l

S

⟩

corresponding to 2

n

trivial (free of crossings)

l

S

dia-grams:

(4)

Once a certain numbering of the crossings is chosen,each one of the 2

n

trivial diagrams can be representedby a binary sequence of the following type:

where the symbol in the

i

th position indicates the typeof the

i

th crossing:

A

or

B

(see the definition of

l

A

and

l

B

diagrams). According to this representation, thesequence

S

determines the state of a diagram. In for-mula (4),

l

S

denotes diagram

l

in the

S

state;

α

(

S

) and

β

(

S

) are the numbers of type

A

and

B

crossings in the

S

state; and the sum is taken over all 2

n

states of dia-gram

l.

A trivial torus diagram contains a certain number ofcircles and, probably, a set of closed nonintersectingcurves wrapping around the torus. Below, these sets arecalled “windings” and are denoted (

m

,

n

), where

m

and

n

are the numbers of intersections of the winding witha meridian and parallel, respectively, on the torus (with-out “meanders”). In order to calculate

⟨

l

S

⟩

, rela-

l⟨ ⟩ a lA⟨ ⟩ b lB⟨ ⟩ ;+=

l ∪⟨ ⟩ c l⟨ ⟩ ;=

⟨ ⟩ 1,=

l⟨ ⟩ aα S( )

bβ S( )

lS⟨ ⟩ .S

∑=

S ABBA…AB ,=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

n

Fig. 2. Planar unit cells.

TECHNICAL PHYSICS LETTERS Vol. 32 No. 5 2006

A POLYNOMIAL INVARIANT OF DOUBLY PERIODIC BRAIDED STRUCTURES 447

tions (2) and (3) must be supplemented by a definitionof the Kauffman bracket ⟨(m, n)⟩> for the given (m, n)winding.

Let us define ⟨(m, n)⟩ so that polynomial (4) does notchange upon torus torsions, that is, is independent ofthe choice of a unit cell for a given 2-braid. The wind-ings possess the following simple properties:

(i) The (m, n) winding contains g = gcd(m, n) com-ponents.

(ii) The number of components of a winding doesnot change upon torus torsion.

(iii) The (m1, n1) winding can be transformed intothe (m2, n2) winding via a sequence of torus torsionsonly provided that gcd(m1, n1) = gcd(m2, n2).

According to the latter property, a necessary condi-tion is that

(5)

The simplest means of satisfying this condition is tointroduce an additional variable t such that ⟨(m, n)⟩ = tg,g = gcd(m, n).

Let us consider a trivial diagram lS containing k cir-cles and set γ(S) = k if the diagram has a winding andγ(S) = k – 1 otherwise. Let g(S) denote the number ofwinding components, and put g(S) = 0 in the absence ofa winding. In these terms, Eq. (4) acquires the follow-ing explicit form:

(6)

Since the values of α, β, and γ remain unchangedupon torsions, the Kauffman bracket (6) is invariant

m1 n1,( )⟨ ⟩ m2 n2,( )⟨ ⟩ ,=

if gcd m1 n1,( ) gcd m2 n2,( ).=

l⟨ ⟩ aα S( )

bβ S( )

cγ S( )

tg S( )

.S

∑=

with respect to the choice of a unit cell. The require-ment that ⟨l⟩ be invariant, as in the classical case, withrespect to the Reidemeister moves Ω2 and Ω3 imposesthe following limitations on the variables a, b, and c:b = a–1 and c = –a2 – a–2 [6, 5]. In order to ensure theinvariance with respect to the Reidemeister move Ω1, itis necessary to pass from ⟨l⟩ to the following poly-nomial:

(7)

where σ(l) is the self-writhe index [6] of diagram l.

Thus, the obtained polynomial of two variables,X(l)(a, t), is an isotopic invariant of unoriented doublyperiodic braids in 3. Particular polynomial invariantscalculated using formula (7) for some simple 2-struc-tres are presented in the table.

The polynomial invariant X(l)(a, t) can be used as abasis for the topological classification of doubly peri-odic braids and for the investigation of structural prop-erties of textiles. Another possible field of application

X l( ) a–( ) 3σ l( )–l⟨ ⟩=

= a–( ) 3σ l( )–a

α S( ) β S( )––a

2a

2––( )

γ S( )t

g S( ),

S

∑

Ω1 Ω2 Ω3

Fig. 3. Reidemeister moves.

Polynomial invariants for some simple braided structures

Type Structure Polynomial invariant

(a) X = a–2 + a2 + 2t2

(b) X = (a4 + a6 – a10)t

(c) X = –(a–6 + a–4 – a–2 – 3 – a2 + a4 + a6)t

(d) X = –(a–14 + a–12 – a–10 – 2a–8 + 2a–4 – 3 – a2 + a4 + a6)t

(e) X = (a–9 – 3a–5 – 2a–3 + 2a–1 +2a – 2a3 – 3a5 +a9)t

(f) X = a–10 – 5a–6 + 3a–2 + 3a2 – 5a6 + a10 + (a–12 – 2a–8 – 3a–4 + 8 – 3a4 – 2a8 + a12)t2

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GRISHANOV et al.

of the proposed invariant is the theory of two-dimen-sional lattice models in statistical physics.

Acknowledgments. The authors are grateful toProf. E.A. Tropp for his useful remarks.

REFERENCES

1. V. F. R. Jones, Pac. J. Math. 137, 311 (1989).

2. L. H. Kauffman, Am. Math. Monthly 95, 195 (1988).

3. F. Y. Wu, Rev. Mod. Phys. 64, 1099 (1992).

4. R. J. Baxter, Exactly Solved Models in StatisticalMechanics (Academic, New York, 1982).

5. V. V. Prasolov and A. B. Sosinskii, Knots, Links, Braids,and 3-Manifolds: An Introduction to the New Invariantsin Low-Dimensional Topology (American MathematicalSociety, Providence, 1997).

6. B. Bollobás, L. Pebody, and D. Weinreich, in Contempo-rary Combinatorics, Bolyai Society Mathematical Stud-ies (Springer, Berlin, 2002), Vol. 10, Chap. 4.

Translated by P. Pozdeev