9 9 Monte Carlo Results for Projected Self-Avoiding 9 1 Polygons: A Two-dimensional Model for Knotted n a Polymers J 7 ] E. Guitter and E. Orlandini h c e m CEA-Saclay, Service de Physique Th´eorique, - t F-91191 Gif-sur-Yvette Cedex, France a t s . February 1, 2008 t a m - d n o c Abstract [ 1 We introduce a two-dimensional lattice model for the description of knot- v ted polymer rings. A polymer configuration is modeled by a closed polygon 1 drawn on the square diagonal lattice, with possible crossings describing pairs 4 0 of strands of polymer passing on top of each other. Each polygon configura- 1 tion can be viewed as the two-dimensional projection of a particular knot. We 0 study numerically the statistics of large polygons with a fixed knot type, us- 9 9 ing a generalization of the BFACF algorithm for self-avoiding walks. This new / algorithm incorporates both the displacement of crossings and the three types t a of Reidemeister transformations preserving the knot topology. Its ergodicity m within a fixed knot type is not proven here rigorously but strong arguments - d in favor of this ergodicity are given together with a tentative sketch of proof. n Assuming this ergodicity, we obtain numerically the following results for the o c statistics of knotted polygons: In the limit of a low crossing fugacity, we find v: a localization along the polygon of all the primary factors forming the knot. i Increasing the crossing fugacity gives rise to a transition from a self-avoiding X walk to a branched polymer behavior. r a 1 The statistical properties of self-avoiding polymers are now very well understood, mainly thanks to the famous equivalence of the problem to a field theory, leading to a natural derivation of scaling laws and critical exponents [1]. In three dimensions however, closed polymers will generally form knots, which, for self-avoiding objects, will restrict the accessible configurations. In the last few years, some progress has been made towards answering some basic questions about knots, showing for instance that a sufficiently long polymer is knotted with probability one [2, 3], or computing the distribution of random knots [4]. However, the statistics of a closed polymer with a fixed knot-type is less well understood. Indeed, the usual field theory approach does not account for this distinction of the knot type, but corresponds rather to a summation over all the possible knot topologies. In particular, the natural question of how the critical exponents depend on the knot topology remains an open issue. Knotted configurations are encountered in the description of closed DNA molecules, with an apparent influence of the knot type on some of their properties [5]. An attempt to understand the role of a fixed knot topology was done in [6, 7, 8, 9, 10, 11] where a numerical study of knotted lattice polygons in the cubic lattice Z3 was performed. There it was shown that, while the connectivity constant and the size exponent ν for the radius of gyration are independent of the knot type, the entropic exponent α for the number of accessible configurations clearly depends on the knot at hand. More precisely, α seems to depend only on the number of primary knots (factors) and increases by one for each added factor in the knot factorization. In practice, all the characterizations of knots involve only their projection in a two-dimensional (2D) plane. Therefore, nothing prevents us from considering a two- dimensional model for knotted polymers. Such a model should be in principle much simpler to study, bothanalytically and numerically. The aim of this paper is precisely to introduce and study numerically a particular model of “projected” polygons on a 2D lattice, describing 2D knotted polymers. In a 2D model for knotted polymers, the self-avoiding constraint in two dimensions is released to allow for what we call “crossings”. By crossings, we actually mean two strands of the chain passing on top of each other in the projection. The object remains self-avoiding in the sense that the underlying and overlying strands cannot be exchanged by passing through each other. With this restriction, each closed polymer has a fixed knot type, which can be preserved in the deformations. Still the object is two-dimensional in the sense that its entropy corresponds to 2D deformations only. In particular, we will recover in some limits some of the usual exponents of 2D self-avoiding walks (SAW). The paper is organized as follows. In the first section, we introduce our lattice model of projected self-avoiding polygons. Section 2 describes the local elementary moves which we use to deform the polygon and explore its accessible phase space at a fixed knot topology. The issue of ergodicity is also discussed in this section where arguments for a proof of ergodicity within a fixed knot type (but not a full proof) are given. The elementary moves are then performed according to a Monte Carlo Metropolis algorithm, drawn to reproduce the most natural grand canonical statistical weight, with a fugacity K per bond of polygon, and a fugacity w per crossing. The updating rules are discussed in section 3. Several improvements of the algorithm are described in section 4. They involve both non local deformations and 2 Figure 1: An example of a PSAP configuration on the square diagonal lattice with six crossings and the topology of a trefoil knot. multiple chains run in parallel. Our numerical results are presented in sections 5, 6 and 7 for w = 1, w 0 and w > 1 respectively. We gather our conclusions in section → 8. 1 The Model We present here the model of Projected Self-Avoiding Polygons (PSAP’s) that we shall use to describe two-dimensional knotted polymers. The model is defined on the square diagonal lattice in two dimensions, namely the simple two dimensional square lattice Z2 completed with the diagonals of the squares. We treat each diagonal as an elementary edge of the lattice, i.e. we consider that the intersection point of the two diagonals is not a vertex of the lattice. The bonds of the polymer can sit on all the edges of the lattice, either vertical, horizontal or diagonal, with at most one bond per lattice edge. Different rules apply for the edges and vertices of the simple square lattice on the one hand and for the diagonal edges on the other hand. We impose a strict self-avoidance at the vertices of the square lattice, i.e. we do not allow two parts of the polymer to either cross each other or even to touch each other at these vertices. We allow crossings to take place inside a square at the intersection point of its two diagonal edges. More precisely, we use the diagonal edges only for crossings, i.e. we impose that a diagonal edge can be occupied if and only if the perpendicular diagonal edge in the same square is also occupied, with the two bonds on these edges forming a crossing. For each pair of occupied diagonal edges inside a square, we distinguish between two different possible crossings according to which of the two bonds of polymer passes on top of the other. We can thus view crossings as a two dimensional projection of a pair of bonds in three dimensions, with one bond 3 I II III Figure 2: The three types (I, II and III) of Reidemeister moves on the projections of knots. Reversing all the crossings also corresponds to allowed transformations. lying on top of the other. Finally, we limit ourselves to closed polymers. Figure 1 shows an example of an allowed configuration with six crossings and the topology of a trefoil knot. Our model can be seen as a simple extension of the usual model describing self- avoiding polymer loopsonthesquarelattice, also referedto asSelf-Avoiding Polygons (SAP). The new ingredient here is the possibility for the polymer, which is strictly self-avoiding on the square lattice, to have crossings taking place on pairs of perpen- dicular diagonal edges inside a square. Again, these “crossings” are viewed as the two dimensional projection of two bonds of polymer passing on top of each other, with thus two distinct allowed configurations according to which of the two bonds is on top of the other. Thanks to this distinction, our model still describes a particular self-avoiding object in the sense that it mimics the projection in two dimensions of a polymer which would be self-avoiding in three dimensions. We will refer to our model as a model of Projected Self-Avoiding Polygons (PSAP’s). Since we use closed poly- gons, a PSAP will in general form knots, i.e. will be the two-dimensional projection of a three dimensional knot. In this sense, we can speak of our model as describing two-dimensional knotted polymers. To fully specify the model and to study the corresponding statistics, we need to assign to each PSAP configuration its weight. Since our aim is the study of knotted polymers of a fixed knot type, we want to attach a non-zero weight only to those configurations which have the desired knot-topology. We will be mainly interested below in rather simple topologies, i.e. that of the unknot ( ), that of the trefoil ∅ knot (3 ), that of the figure eight knot (4 ) and that of the composite knots made 1 1 of two trefoils (3 #3 ) or a trefoil and a figure eight (3 #4 ). In practice, we will 1 1 1 1 start with an initial configuration fixing the knot type and we will explore the phase spaceaccessiblebyperformingsuccessive transformations(moves) onthePSAPwhich preserve its topology. Aswell known fromknot theory, localdeformations exist onthe projection of a knot which preserve its topology. These deformations are classified 4 as the Reidemeister moves [12], which are of three types as described in figure 2. These topological deformations are sufficient to pass from any two configurations of the same knot type. The set of allowed moves in our lattice model will be described in detail in the next section. For all the accessible conformations of PSAP of a given knot type, we moreover weigh the configuration by a factor nQKnwc π( ) = (1) P G (Q,K,w) τ where n is the number of bonds of the PSAP configuration , with a fugacity K P per bond, and c is the number of crossings, with a fugacity w per crossing. We also artificially introduce a factor nQ with a factor Q > 1 for numerical convenience. This factor is unimportant for averages at fixed n (canonical ensemble) but it can improve the statistics in a grand-canonical ensemble with varying n. We typically take Q = 2 in the following. The denominator in eq.(1) is the grand-canonical partition function at fixed knot-topology τ, which normalizes the weights so that the total weight for all accessible configurations is equal to unity. For the knot-topology preserving moves, the above weight will dictate the probability to accept or reject the deformation. 2 Local Elementary Moves In this section, we describe the local part of our algorithm, which is a grand-canonical implementation, since it involves changes in the number of bonds of the projected polygon. Our algorithmuses local elementary moves of four different types depending on how many elementary squares of Z2 (plaquettes) are involved in the move. We will use single, double, 3-plaquette (or corner) and 6-plaquette moves. We first give here a description of all these elementary moves. The way we decide in the algorithm which type of move we attempt and with which probability we accept or reject the attempted move is described in the next section. Single Moves Single Moves are performed on a single plaquette, and may involve length changes through the addition or subtraction of bonds in the PSAP. This moves are defined as follows: given a horizontal or vertical bond (p ,p ) of the i i+1 current PSAP, we pick a unit vector e perpendicular to (p ,p ). This defines i i i+1 a plaquette (p ,p ,p +e ,p +e ). A move can occur only if the edge (p + i i+1 i+1 i i i i e ,p +e )isnotoccupied. Themovedepends onhowmanyedgesareoccupied i i+1 i on the plaquette. If exactly two edges are occupied, we exchange occupied and unoccupied edges. This corresponds to a on bead flip (transformation S in figure 3). If only one edge is occupied, we shift (p ,p ) one lattice I i i+1 spacing in the direction of e and complete the PSAP by two additional bonds, i either on the sides, creating a kink (transformation S in figure 3), or on II the diagonals, creating a crossing (transformation S or S′ in figure 4). If IV IV exactly three edges are occupied, we move (p ,p ) to (p + e ,p + e ) and i i+1 i i i+1 i delete the two other bonds. The result is either a kink deletion (transformation S in figure 3) or a crossing deletion (transformation S or S′ in figure 4). III V V 5 t t t - t t t S I S t t II - (cid:27) t t t t S III Figure 3: Single local c-preserving moves. (S ) One-bead flip (∆n = 0). (S ) Kink I II insertion (∆n = +2). (S ) Kink deletion (∆n = 2). III − S t t IV- (cid:0) (cid:27) (cid:0) t t t(cid:0) t S V 2 4 S S S′ t t IV- @ (cid:27) @ t t t @t S′ V 2 4 S S Figure 4: Single local c-changing moves: Reidemeister I moves. (S ,S′ ) Crossing IV IV insertion (∆c = +1). (S ,S′ ) Crossing deletion (∆c = 1). V V − The single moves can be classified into two different sub-groups depending on whether they do or do not preserve the number of crossings c of the PSAP. The c-preserving moves illustrated in figure 3 are known in the literature as the BFACF moves [13, 14, 15]. While the move S (one-bead flip) is n-preserving, I the moves S (kink insertion) and S (kink deletion) change the length of II III the PSAP respectively by ∆n = +2 and ∆n = 2. The c-changing moves are − illustrated in figure 4. In this case, in addition to the change ∆n = 2 in the ± length of the PSAP, the number of crossings changes by an amount ∆c = +1 (moves S and S′ ) or ∆c = 1 (moves S and S′ ). The moves S′ and S′ IV IV − V V IV V are identical to the moves S and S apart from the reversing of the crossing IV V between and . All these c-changing moves correspond to a Reidemeister I 4 4 S S transformation. Double Moves A double move is performed on two adjacent plaquettes. Double moves are selected by first choosing, along the current PSAP, a vertex p . Dif- i ferent kinds of double moves are considered, depending on the relative orienta- tions of the two bonds (p ,p ) and (p ,p ) shared by the vertex p (see figure i−1 i i i+1 i 5): 6 p p i−1 i+1 t t @ (cid:0) pi−t1 tpi tpi+1 @@p(cid:0)ti(cid:0) (a) (b) p i+1 t (cid:0) p p (cid:0) i−t1 it(cid:0) (c) Figure 5: Different cases considered for double moves: (a) parallel vertical or hori- zontal bonds; (b) perpendicular diagonal bonds; (c) bonds at 135◦. Case of parallel vertical or horizontal bonds: If the two bonds are par- allel, we choose one of the two possible unit vectors e perpendicular to i (p ,p ), and we check if the vertex p = p +e belongs to the current i−1 i+1 j i i PSAP configuration. If it does, we look for a 2-plaquette configuration such as that depicted in figure 6. If this 2-plaquette configuration is 1 D encountered, we make the local transformation D by exchanging p and I i p = p + e , leading to the configuration with two more crossings, or j i i 2 D make the similar transformation D′ leading to the reversed configuration I . 2 D Case of perpendicular diagonal bonds: If (p ,p ) and (p ,p ) are on i−1 i i i+1 perpendicular diagonal edges, the PSAP necessarily has two crossings in- volving two strands of polygon. We check that we have a configuration of type or i.e. that the two strands are not entangled (see figure 6). If 2 2 D D so, we then make the local transformation D or D′ to suppress the two II II crossings. Case of bonds at 135◦: If one of the bonds, say (p ,p ) is vertical or hor- i−1 i izontal, and the other bond (p ,p ) is at 135◦ on a diagonal edge, the i i+1 procedure is analogous to the case of parallel bonds, with e the unit vec- i tor perpendicular to (p ,p ) and inside the convex sector (p ,p ,p ). i−1 i i−1 i i+1 After checking that the vertex p = p +e belongs to the current PSAP, j i i we look for a 2-plaquette configuration such as the one depicted in figure 7. We then let the crossing diffuse one step, according to transformation D III of figure 7. A similar procedure is used for the configuration obtained 3 D by reversing the crossing of . 3 D Finally, no double move is performed if the two consecutive bonds shared by p i belong to the same plaquette. In this case a corner move (see below) will be attempted instead. Transformations D and D can be though of as Reidemeister II moves while I II transformation D is simply a diffusion of the crossing to a neighboring pla- III 7 t t t D t t t I - (cid:0)@ (cid:27) (cid:0) @ t t t t(cid:0) t @t D II 1 2 D D Figure 6: Double c-changing moves : Reidemeister II moves. t t t D t t t III @ - @ @ @ t t @t t @t t 3 D Figure 7: Double c-preserving move: diffusion of a crossing. quette. In addition to the configurations described above, there are also 2-plaquette configurations in which the two involved strands of the PSAP are consecutive. We call these configurations degenerate since can be transformed one into an- other by the same set of double moves above. Some of such configurations are illustrated in figure 8. Corner Moves . In these moves, three plaquettes of the underlying squared lattice are involved. As for the double moves, we first choose a vertex p of the current i PSAP. A corner move can be performed if the bonds (p ,p ) and (p ,p ), i−1 i i i+1 shared by the chosen vertex p , are on the two consecutive edges of a square, i as illustrated in figure 9. In this case, we check if the vertex p = p + j i − p +p . belongs to the current PSAP. If so, we look for local configurations i+1 i−1 involving three plaquettes such as those depicted in figure 10 ( , ) and in 1 2 C C figure 11 ( ). If the local 3-plaquette configuration does not correspond to any 3 C of such configurations no move is performed. Otherwise, we attempt one of the corner moves illustrated in figures 10 and 11. We can think of the corner t t t D t t t I - (cid:0)@ (cid:27) (cid:0) @ t t t t(cid:0) t @t D II 1 2 D D t t t D t t t III @ - @ @ @ t t @t t @t t 3 D Figure 8: Degenerate configurations for double moves. 8 p p i i+1 t t t 3 p p i−1 j Figure 9: Starting configuration for a corner move. p C p i II i t t t t t t (cid:27) (cid:0) - (cid:0) t tpj t C t (cid:0)t(cid:0)pj t I (cid:0) t t t(cid:0) t 1 2 C C Figure 10: Corner c-changing move: Reidemeister II move around a corner. moves C and C as Reidemeister II moves, whereas the corner move of figure I II 11 corresponds to a diffusion of a crossing through the corner. Note that this diffusion around a corner also involves a rotation of the the crossing on the square lattice. Corner moves C′ and C′ connecting configuration to are I II C1 C2 also possible. In addition, as for the double moves, there are again degenerate configurations that can be transformed one into another by the same corner moves described above. Some of them are illustrated in figure 12. 6-plaquette Moves (Reidemeister III moves). This moves are performed by choos- ing a diagonal bond in the current PSAP. This bond singles out a crossing. We then look for the presence of two neighboring crossings along two adjacent diagonal directions, chosen at random. We finally look for configurations such as the one depicted in figure 13. The performed move R corresponds to a I Reidemeister III move. Here again degenerate configurations exist which can be changed one into another. Note that double, corner and 6-plaquette moves do not change the number n of bonds in the PSAP. p p i i t t t t t t (cid:0) (cid:0) - t t(cid:0) t t t t pj CIII @@ pj t t t @t 3 C Figure 11: Corner c-preserving move : diffusion of a crossing around a corner. 9 p C p i II i t t t t t t (cid:27) (cid:0) - (cid:0) t tpj t C t (cid:0)t(cid:0)pj t I (cid:0) t t t(cid:0) t 1 2 C C p p i i t t t t t t (cid:0) (cid:0) - t t(cid:0) t t t t pj CIII @@ pj t t t @t 3 C Figure 12: Degenerate cases of corner moves. t t t t t t t t (cid:0) @ (cid:0) (cid:0) @ (cid:0) t(cid:0) t t @t - t t(cid:0) t t (cid:0) @ (cid:0) (cid:0) R @ (cid:0) t t(cid:0) t t I t @t t(cid:0) t 1 R Figure 13: 6-plaquette c-preserving move : Reidemeister III move. To end this section let us mention that: For all the moves drawn above, equivalent moves exist which are obtained by • 90◦ rotations or mirror reflections. If after our checks we do not find one of the above described environments, no • move is performed. As described in the next section we have an exhaustive procedure to decide • which type of move will be attempted. All the moves above clearly preserve the knot-topology. Therefore any possible deformation will maintain the PSAP inside the set of conformations having the same knot type as the conformation one started with. In the following, we will assume that the algorithm is ergodic, although a full proof of ergodicity goes far beyond our goal. By ergodicity, we mean that starting from a given conformation, any other conformation with the same knot-topology can be obtained by a series of successive elementary moves. In the absence of crossings, our algorithm reduces to the BFACF algorithm [13, 14, 15], which is known to be ergodic. More precisely, the BFACF moves (moves S , S and S ) allow to deform any configuration of a set of self- I II III avoiding paths onto any other configuration with the same topology and, if some of the paths are open, with the same positions of the end-points. Our algorithm completes the BFACF algorithm with knot-topology preserving moves allowing the 10