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 Tekhnicheskoœ Fiziki, 2006, Vol. 32, No. 10, pp. 61–67.
A Polynomial Invariant of Doubly Periodic Braided Structures S. A. Grishanov, V. R. Meshkov, and A. V. Omel’chenko De Monfort University, Leicester, United Kingdom St. Petersburg State University of Technology and Design, St. Petersburg, 191065 Russia St. Petersburg State Technical University, St. Petersburg, 195251 Russia e-mail:
[email protected] Received November 28, 2005
Abstract—A new isotopic Kauffman-type polynomial invariant of two variables for doubly periodic braided structures is constructed. PACS numbers: 04.20.Jz DOI: 10.1134/S1063785006050221
For a long time, the properties of spatial configurations with knots and entanglements received only the interest of mathematicians. Knot theory, being to a considerable extent of “physical” origin, has grown into an independent direction in topology. However, recent decades have revealed the deep relations of this theory with various fields in physics. In particular, it was established that the Young–Baxter equation, playing an important role in statistical physics and quantum theory, also appears in knot theory, where it is related to one of the so-called Reidemeister moves—transforms which do not change the topological types of knots. The establishment of this relationship led to the discovery of new powerful invariants in knot theory, such as polynomials of the Jones, Kauffman, and HOMFLY types [1–3]. At the same time, the physics received new analytical methods. Doubly periodic braided structures (referred to below as 2-braids), as well as knots and entanglements, are frequently encountered in applications. In particular, two-dimensional models in statistical physics are considered on such lattices [4] and numerous examples of “real” 2-braids are offered by textiles. Investigation into the topological properties of 2-braids is of considerable interest for such applications. Despite this significance, 2-braids have not been specially considered as topological objects so far and, in particular, no classification of 2-braids has been developed.
the isotopic deformations that retain periodicity of the structure. A 2-braid (and any invariant of this structure) is fully determined once the minimum repeat element— the unit cell—is defined. Such an element can be chosen in an infinite number of ways; an isotopic invariant of the given structure must be independent of the selection of a unit cell. This Letter presents a new polynomial invariant for 2-braids, which obeys this condition. Knot diagrams. Knot invariants are conveniently determined using diagrams representing nondegenerate planar projections with indications of the types of crossings [5]. A 2-braid is naturally brought into correspondence with a diagram on a torus. Such a diagram is obtained by identifying the opposite sides of a unit cell chosen on a flat diagram of the given structure (Fig. 1). Apparently, this correspondence is not unique, as is illustrated in Fig. 2, since there is a one-to-one correspondence between the manifold of reduced fractions ±p/q and the manifold of unit cells (defined to within a shift). The relation of equivalence on a manifold of torus diagrams corresponding to the given 2-braid can be set in terms of torus torsions, which are defined as follows. Let us cut the torus along an arbitrary meridian, rotate one edge of the cut through 360° in its plane, and con-
The main tool for the investigation of knots and entanglements are invariants. The invariant is a function defined on a set of knots, which has the same values for equivalent knots. Two knots are considered topologically equivalent (isotopic) provided that one of them can be continuously transformed into another without self-intersections (such deformation is called isotopy). In the case of 2-braids, it is natural to consider 445
Fig. 1. Diagrams of knots, entanglements, and 2-braids.
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of torus and a finite number of the Reidemeister moves and torus torsions. Thus, any function defined on a manifold of torus diagrams, which is not changed by torsions and Reidemeister moves, determines an isotopic invariant of 2-braids. Below, we construct such a function with values in the manifold of polynomials. Isotopic invariant. Following Kauffman [2], let us bring every torus diagram l into correspondence with a polynomial 〈l 〉 of variables a, b, and c, which obeys the following relations: 〈 l〉 = a 〈 l A〉 + b 〈 l B〉 ;
(1)
〈 l ∪ 〉 = c 〈 l〉 ;
(2)
〈 〉 = 1,
(3)
Fig. 2. Planar unit cells.
nect the edges. This procedure will be referred to as the meridian torsion. By the same token, we can define the torsion along a parallel. Obviously, using such meridian and parallel torsions, it is possible to transform an arbitrary unit cell into any other unit cell.
where
Equivalent knots. Two knots (or entanglements) are called equivalent if there is a family of diffeomor3 smoothly dependent on t ∈ phic maps ft: 3 [0, 1], in which f0 is the identical diffeomorphism and f1 maps K1 into K2 (so-called isotopy relating K0 and K1). It should be noted that the requirement of the one-toone correspondence of ft maps prohibits the self-intersection of a knot in the course of isotopy. The concept of a planar isotopic mapping for the knot diagrams is formulated similarly.
are three diagrams that are identical outside the dashed circle and is the circle diagram. Polynomial 〈l 〉 is called the Kauffman bracket [5]. Let n be the number of crossings in diagram l. By consequently applying relation (1) to each crossing, it is possible to express 〈l 〉 in terms of polynomials 〈lS 〉 corresponding to 2n trivial (free of crossings) lS diagrams:
In contrast to the case of usual knot diagrams, the equivalence of 2-braids represented by two torus diagrams cannot be unambiguously judged using only the Reidemeister moves and isotopies. Indeed, these diagrams may correspond to two different unit cells of the same structure. In order to eliminate the problem related to the ambiguity of the choice of a unit cell, the possible transformations of torus diagrams must be supplemented by the torsions defined above. Then, the following generalization of the Reidemeister theorem is valid. Theorem. Two torus diagrams correspond to isotopic 2-braids if and only if these diagrams can be transformed into one another using isotopies on the surface
〈 l〉 =
; lA =
∑a
; lB =
α(S) β(S)
b
〈 l S〉 .
(4)
S
Once a certain numbering of the crossings is chosen, each one of the 2n trivial diagrams can be represented by a binary sequence of the following type: S =
ABBA…AB
,
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
In terms of the knot diagrams, two unoriented knots K1 and K2 are called equivalent if and only if the K1 diagram can be transformed into the K2 diagram using planar isotopies and the Reidemeister moves Ω1 , Ω2 , and Ω3 (Reidemeister theorem). For torus diagrams, the Reidemeister moves Ω1 , Ω2 , and Ω3 can be defined in the same manner as for the planar diagrams, and the isotopy on the torus surface can be introduced by analogy to the planar isotopy.
l=
n
where the symbol in the ith position indicates the type of the ith crossing: A or B (see the definition of lA and lB diagrams). According to this representation, the sequence S determines the state of a diagram. In formula (4), lS 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 2n states of diagram l. A trivial torus diagram contains a certain number of circles and, probably, a set of closed nonintersecting curves wrapping around the torus. Below, these sets are called “windings” and are denoted (m, n), where m and n are the numbers of intersections of the winding with a meridian and parallel, respectively, on the torus (without “meanders”). In order to calculate 〈lS 〉, relaTECHNICAL PHYSICS LETTERS
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A POLYNOMIAL INVARIANT OF DOUBLY PERIODIC BRAIDED STRUCTURES
tions (2) and (3) must be supplemented by a definition of the Kauffman bracket 〈(m, n)〉> for the given (m, n) winding. Let us define 〈(m, n)〉 so that polynomial (4) does not change upon torus torsions, that is, is independent of the choice of a unit cell for a given 2-braid. The windings possess the following simple properties: (i) The (m, n) winding contains g = gcd(m, n) components. (ii) The number of components of a winding does not change upon torus torsion. (iii) The (m1, n1) winding can be transformed into the (m2, n2) winding via a sequence of torus torsions only provided that gcd(m1, n1) = gcd(m2, n2). According to the latter property, a necessary condition is that 〈 ( m 1, n 1 )〉 = 〈 ( m 2, n 2 )〉 , if
The simplest means of satisfying this condition is to introduce an additional variable t such that 〈(m, n)〉 = t g, g = gcd(m, n). Let us consider a trivial diagram lS containing k circles and set γ(S) = k if the diagram has a winding and γ(S) = k – 1 otherwise. Let g(S) denote the number of winding components, and put g(S) = 0 in the absence of a winding. In these terms, Eq. (4) acquires the following explicit form:
∑a
α(S) β(S) γ (S) g(S)
b
c
t
(6)
.
S
Since the values of α, β, and γ remain unchanged upon torsions, the Kauffman bracket (6) is invariant
Ω2
Ω3
Fig. 3. Reidemeister moves.
with respect to the choice of a unit cell. The requirement that 〈l〉 be invariant, as in the classical case, with respect to the Reidemeister moves Ω2 and Ω3 imposes the following limitations on the variables a, b, and c: b = a–1 and c = –a2 – a–2 [6, 5]. In order to ensure the invariance with respect to the Reidemeister move Ω1 , it is necessary to pass from 〈l〉 to the following polynomial:
(5)
gcd ( m 1, n 1 ) = gcd ( m 2, n 2 ).
〈 l〉 =
Ω1
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X ( l ) = ( –a ) = ( –a )
– 3σ ( l )
∑a
– 3σ ( l )
α(S) – β(S)
2
〈 l〉 –2 γ ( S ) g ( S )
( –a – a )
t
,
S
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 doubly periodic braids in 3. Particular polynomial invariants calculated using formula (7) for some simple 2-structres are presented in the table. The polynomial invariant X(l)(a, t) can be used as a basis for the topological classification of doubly periodic braids and for the investigation of structural properties of textiles. Another possible field of application
Polynomial invariants for some simple braided structures Type
Structure
Polynomial invariant
(a)
X = a–2 + a2 + 2t 2
(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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of the proposed invariant is the theory of two-dimensional lattice models in statistical physics. Acknowledgments. The authors are grateful to Prof. E.A. Tropp for his useful remarks. REFERENCES 1. V. F. R. Jones, Pac. J. Math. 137, 311 (1989).
4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982). 5. V. V. Prasolov and A. B. Sosinskii, Knots, Links, Braids, and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology (American Mathematical Society, Providence, 1997). 6. B. Bollobás, L. Pebody, and D. Weinreich, in Contemporary Combinatorics, Bolyai Society Mathematical Studies (Springer, Berlin, 2002), Vol. 10, Chap. 4.
2. L. H. Kauffman, Am. Math. Monthly 95, 195 (1988).
Translated by P. Pozdeev
3. F. Y. Wu, Rev. Mod. Phys. 64, 1099 (1992).
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