hamiltonian formalism for nonlocal lagrangians a hamiltonian formalism is set up for nonlocal lagrangian systems the method is based on obtaining an equivalent singular first order lagrangian which is processed according to the standard legendre transformation and then the resulting hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints finally the standard results for local lagrangians of any order are obtained as a particular case

irreversibility implies the occurrence of nonmonotonic power spectra both continuous time and discrete time systems driven by l2 exponential ergodic stationary markov processes are investigated we prove that in both cases irreversibility implies the occurrence of nonmonotonic power spectra of some observables this result combined with the work of jiang and zhang j math phys 44 4681 4689 2003 tells us that in the continuous time case reversibility can be characterized by monotonicity in all power spectra while this is not true in the discrete time case

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

a monte carlo algorithm for lattice ribbons a lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions the ribbon can be closed to form an object which is topologically either a cylinder or a mobius band depending on whether its surface is orientable or nonorientable we describe a grand canonical monte carlo algorithm for generating a sample of these ribbons prove that the associated markov chain is ergodic and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons

path integral calculation of the writhe for circular semiflexible polymers earlier obtained numerical and analytical non path integral results for the writhe of a circular polymer are reproduced exactly employing the available path integral models for the semiflexible chains it is confirmed analytically that the average writhe scales with the root of the chain length we also briefly discuss related topological problems e g knotting of circular polymers which could be formulated and solved with the use of path integrals

the abundance of unknots in various models of polymer loops a veritable zoo of different knots is seen in the ensemble of looped polymer chains whether created computationally or observed in vitro at short loop lengths the spectrum of knots is dominated by the trivial knot unknot the fractional abundance of this topological state in the ensemble of all conformations of the loop of n segments follows a decaying exponential form exp n n0 where n0 marks the crossover from a mostly unknotted i e topologically simple to a mostly knotted i e topologically complex ensemble in the present work we use computational simulation to look closer into the variation of n0 for a variety of polymer models among models examined n0 is smallest about 240 for the model with all segments of the same length it is somewhat larger 305 for gaussian distributed segments and can be very large up to many thousands when the segment length distribution has a fat power law tail

distribution of the distance between opposite nodes of random polygons with a fixed knot we examine numerically the distribution function fk r of the distance r between opposite polygonal nodes for random polygons of n nodes with a fixed knot type k here we consider three knots such as 31 and 3131 in a wide range of r the shape of fk r is well fitted by the scaling form 1 of self avoiding walks the fit yields the gaussian exponents and k 1 furthermore if we re scale the intersegment distance r by the average size rk of random polygons of knot k the distribution function of the variable r rk should become the same gaussian distribution for any large value of n and any knot k we also introduce a fitting formula to the distribution gk r of gyration radius r for random polygons under some topological constraint k

time asymmetric fokker type action and relativistic wave equations a relativistic two particle system with an arbitrary linear combination of scalar and vector time asymmetric fokker type interactions in a two dimensional spacetime is considered within the framework of the front form of dynamics it is shown that the corresponding mass shell equation takes the form of a linear relation between the generators of the lie algebra so 2 1 an algebraic quantization of the system is proposed and a closed form for the mass spectrum is obtained the relativistic wave equation obtained by barut and rasmussen for the h atom is generalized to the case of an arbitrary linear combination of scalar and vector interactions an extension of the results to the system in four dimensional spacetime is suggested

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

knottedness in self avoiding walks in this letter the author explores the knottedness of self avoiding walks he was not able to represent the data as a power law

knot probability for lattice polygons in confined geometries we study the knot probability of polygons confined to slabs or prisms considered as subsets of the simple cubic lattice we show rigorously that almost all sufficiently long polygons in a slab are knotted and we use monte carlo methods to investigate the behaviour of the knot probability as a function of the width of the slab or prism and the number of edges in the polygon in addition we consider the effect of solvent quality on the knot probability in these confined geometries

poland scheraga models and the dna denaturation transition poland scheraga models were introduced to describe the dna denaturation transition we give a rigorous and refined discussion of a family of these models we derive possible scaling functions in the neighborhood of the phase transition point and review common examples we introduce a self avoiding poland scheraga model displaying a first order phase transition in two and three dimensions we also discuss exactly solvable directed examples this complements recent suggestions as to how the poland scheraga class might be extended in order to display a first order transition which is observed experimentally

slow topological time scale of knotted polymers we investigate the rouse dynamics of a flexible ring polymer with a prime knot within a monte carlo approach we locate the knot follow its diffusion and observe the fluctuations of its length a topological time scale that is slower than the rouse one is found to determine the long time behavior of several dynamical quantities the value of the associated dynamical exponent is zt 2 32 0 1

knot localization in adsorbing polymer rings we study by monte carlo simulations a model of a knotted polymer ring adsorbing onto an impenetrable attractive wall the polymer is described by a self avoiding polygon on the cubic lattice we find that the adsorption transition temperature the crossover exponent ensuremath phi and the metric exponent ensuremath nu are the same as in the model where the topology of the ring is unrestricted by measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized this knot localization transition is triggered by the adsorption transition but is accompanied by a less sharp variation of the exponent related to the degree of localization indeed for a whole interval below the adsorption transition one can not exclude a contiuous variation with temperature of this exponent deep into the adsorbed phase we are able to verify that knot localization is strong and well described in terms of the flat knot model

critical exponents for square lattice trails with a fixed number of vertices of degree 4 we prove several previously conjectured results about the number of n edge trails and n edge embeddings of eulerian graphs each with a fixed number k of degree 4 vertices in the lattice 2 in particular under the assumption that the relevant critical exponents exist we prove that the difference between the critical exponent for closed trails eulerian graph embeddings and that for self avoiding circuits polygons is exactly k the number of degree 4 vertices similarly we prove that the difference between the critical exponent for either open trails or open eulerian graph embeddings and that for self avoiding walks is also k these results are proved by establishing upper and lower bounds for the number of n edge embeddings of closed open eulerian graphs with k vertices of degree 4 in terms of the number of n edge self avoiding polygons walks the lower bounds are proved using a kesten pattern theorem argument and the upper bounds are established by developing based on a detailed case analysis a method for removing vertices of degree 4 from an embedding by altering at most a constant independent of n number of vertices and edges of the embedding the work presented here extends and improves the arguments first given in the work of zhao and lookman 1993 j phys a math gen 26 1067 76

automatic classification of protein structure by using gauss integrals we introduce a method of looking at analyzing and comparing protein structures the topology of a protein is captured by 30 numbers inspired by vassiliev knot invariants to illustrate the simplicity and power of this topological approach we construct a measure scaled gauss metric sgm of similarity of protein shapes under this metric protein chains naturally separate into fold clusters we use sgm to construct an automatic classification procedure for the cath2 4 database the method is very fast because it requires neither alignment of the chains nor any chain chain comparison it also has only one adjustable parameter we assign 95 51 of the chains into the proper c class a architecture t topology and h homologous superfamily fold find all new folds and detect no false geometric positives using the sgm we display a map of the space of folds projected onto two dimensions show the relative locations of the major structural classes and zoom into the space of proteins to show architecture topology and fold clusters the existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification

detecting knots in self avoiding walks in this comment the authors discuss problems in the method recently proposed by windwer ibid vol 22 p l605 1989 for computing knot probability in self avoiding walks on the simple cubic lattice

loose flat knots in collapsed polymers we consider single ring polymers which are confined on a plane but maintain a fixed three dimensional knotted topology the equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by n step polygons with a number of self intersections restricted to the minimum compatible with the topology this allows to define the size s of the flat knots and to study their localization properties due to the presence of both excluded volume and attractive interactions the model undergoes a theta transition accurate monte carlo results show that while in the high temperature swollen regime both prime and composite knot components are localized s n n t with t 0 in the low temperature compact phase they are fully delocalized t 1 right at the theta transition weak localization prevails t 0 44 0 02 part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations and by applying the scaling theory of polymer networks of fixed topology in particular t 3 7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

a fractional dirac equation and its solution this paper presents a fractional dirac equation and its solution the fractional dirac equation may be obtained using a fractional variational principle and a fractional klein gordon equation both methods are considered here we extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of caputo derivatives by applying the variational principle to a fractional action s we obtain the fractional euler lagrange equations of motion we present a lagrangian and a hamiltonian for the fractional dirac equation of order we also use a fractional klein gordon equation to obtain the fractional dirac equation which is the same as that obtained using the fractional variational principle eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard dirac equation we also provide expressions for the path integral quantization for the fractional dirac field which in the limit 1 approaches to the path integral for the regular dirac field it is hoped that the fractional dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

torsion of polygons in the torsion of polygons and self avoiding walks in the cubic lattice is a measure of the self entanglement of these objects we consider several definitions of torsion in polygons and introduce a fugacity conjugate to the torsion in our models we study the thermodynamic behaviour of these models using probabilistic methods and rigorous methods from statistical mechanics in particular we prove that at least one of our models has a non analyticity in its free energy corresponding to a transition between phases with high and low torsion

lattice stick number of knots the minimal number of straight line segments required to construct a polygonal presentation of the knot k in the cubic lattice is called the lattice stick number of the knot k denoted by it is known that if the crossing number of k satisfies and the main result of this paper is to improve this to if furthermore we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers

new pattern theorems for square lattice self avoiding walks and self avoiding polygons a general pattern theorem for weighted self avoiding polygons saps and self avoiding walks saws in is obtained the pattern theorem for saps fits into the general framework of the pattern theorem for lattice clusters introduced by madras 1999 ann comb 3 357 84 note that unlike other pattern theorems proved for saps this pattern theorem does not rely on first establishing a relationship between saps and saws these results are applied to obtain pattern theorems for self interacting saps and self interacting saws

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

the distribution of alexander polynomials of knots confined to a thin layer numerical hammagraphy is used to determine the statistical distribution of knots which are confined to a thin layer the statistics used are based on more than 2 106 knots among various striking features is a marked regularity in the occurrence of the prime knots

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

the number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

fractional hamiltonian analysis of higher order derivatives systems the fractional hamiltonian analysis of 1 1 dimensional field theory is investigated and the fractional ostrogradski s formulation is obtained the fractional path integral of both simple harmonic oscillator with an acceleration squares part and a damped oscillator are analyzed the classical results are obtained when fractional derivatives are replaced with the integer order derivatives

numerical upper bounds on ropelengths of large physical knots numerical computations explored for this paper show that an upper bound on the ropelength of large knots with crossing number n grows as fast as nln2n the algorithms to randomly generate samples of such large knots and to determine an upper bound on the ropelength for each knot are described the numeric results are presented and compared to the smallest known theoretical upper bounds on ropelength

statistical topology and knotting of fluctuating filaments abstract the aim of these notes is to provide an introduction to the topic of statistical topology with this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments some questions that we are going to address are the following i what is a knot and how can we identify it ii which is the probability of finding a random curve that is knotted iii how complex are these knots iv how big are they to try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some state of the art theoretical and numerical techniques exercises and lists of open problems will be provided too

the dimensions of knotted polygons the authors study the dimensions mean square radius of gyration and mean span of self avoiding polygons on the simple cubic lattice with fixed knot type the approach used is a monte carlo algorithm which is a combination of the bfacf algorithm and the pivot algorithm so that the polygons are studied in the grand canonical ensemble but the autocorrelation time is not too large they show that although the dimensions of polygons are sensitive to knot type the critical exponent v and the leading amplitude are independent of the knot type of the polygon the knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour

entropic exponents of lattice polygons with specified knot type ring polymers in three dimensions can be knotted and the dependence of their critical behaviour on knot type is an open question we study this problem for polygons on the simple cubic lattice using a novel grand canonical monte carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon we conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

treatment of higher order lagrangians via the construction of dynamically equivalent first order lagrangians for a given in general singular lagrangian containing higher order time derivatives a dynamically equivalent lagrangian with only first order time derivatives is constructed a hamiltonian structure for this first order lagrangian is then found with the use of the dirac theory of constraints it is shown that in the case of a nonsingular higher order lagrangian the ostrogradsky dynamics is derived in this way further it is shown that ambiguities characteristic of higher order lagrangian systems do not appear when using this construction in particular it is shown that the addition of a total time derivative term to the higher order lagrangian can only induce a time independent canonical transformation even in the case of a singular lagrangian

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

the writhe of polygons on the face centred cubic lattice for polygons on the simple cubic lattice there is an important theorem due to lacher and sumners which shows that the writhe of a polygon is the average of the linking numbers of the polygon and its pushoffs in four particular directions this implies that the writhe of a polygon on the simple cubic lattice is always rational we prove a related theorem for the face centred cubic fcc lattice but show that on this lattice the probability that a polygon of length n has irrational writhe tends to unity as n tends to infinity in addition we show that the expectation of the absolute value of the writhe increases at least as fast as n 1 2 for large n

on the imaginary real ratio rule of power spectra the power spectra of real observables for continuous time systems driven by l2 exponential ergodic stationary markov processes are investigated we find the necessary and sufficient condition for the power spectra to be monotonic nonmonotonic which includes the famous imaginary real ratio rule as a special case we also give some examples as applications

what is the length of a knot in a polymer we give statistical definitions of the length l of a loose prime knot tied into a long fluctuating ring macromolecule monte carlo results for the equilibrium good solvent regime show that l nt where n is the ring length and t 0 75 is independent of the knot type in the collapsed regime below the theta temperature length determinations based on the entropic competition of different knots within the same ring show knot delocalization t 1

bounds for the minimum step number of knots confined to slabs in the simple cubic lattice volume confinement is a key determinant of the topology and geometry of a polymer however the direct relationship between the two is not fully understood for instance recent experimental studies have constructed p4 cosmids i e p4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their dna knot distribution differs from that of wild type bacteriophage p4 in particular it was observed that the complexity of the knots decreases sharply with the length of the packed genome this problem is the motivation of this paper here a polymer is modeled as a self avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes i e bounded within a slab we estimate the minimum length required for such a polygon to realize a knot type our numerical simulations show that in order to realize a prime knot with up to ten crossings in a 1 slab i e a slab of height 1 one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition in the case of the trefoil knot we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1 slab is 26 which is greater than 24 the known minimum length required to tie a trefoil without a confinement condition additionally we find that in the 1 slab not all geometrical realizations of a given knot type are equivalent under bfacf moves this suggests that in certain confined volumes knowing the topology of a polymer is not enough to describe all its states

upper bounds on the minimum length of cubic lattice knots knots have been considered to be useful models for simulating molecular chains such as dna and proteins one quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot in this paper we consider every knot in the cubic lattice in particular the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice diao introduced this term he used minimum edge number instead and proved that the minimum length of the trefoil knot 31 is 24 also the minimum lengths of the knots 41 and 51 are known to be 30 and 34 respectively in this paper we find a general upper bound of the minimum length of a nontrivial knot k except the trefoil knot in terms of the minimum crossing number c k the upper bound is moreover if k is a non alternating prime knot then the upper bound is our work can be considered a direct consequence of the results obtained by the authors in hong et al 2012 arxiv 1209 0048 furthermore if k is n 1 n torus knot then the upper bound is

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

scaling behavior of random knots using numerical simulations we investigate how overall dimensions of random knots scale with their length we demonstrate that when closed non self avoiding random trajectories are divided into groups consisting of individual knot types then each such group shows the scaling exponent of 0 588 that is typical for self avoiding walks however when all generated knots are grouped together their scaling exponent becomes equal to 0 5 as in non self avoiding random walks we explain here this apparent paradox we introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots we also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

monte carlo results for projected self avoiding polygons a two dimensional model for knotted polymers we introduce a two dimensional lattice model for the description of knotted polymer rings a polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice with possible crossings describing pairs of strands of polymer passing on top of each other each polygon configuration can be viewed as the two dimensional projection of a particular knot we study numerically the statistics of large polygons with a fixed knot type using a generalization of the bfacf algorithm for self avoiding walks this new algorithm incorporates both the displacement of crossings and the three types of reidemeister transformations preserving the knot topology its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof assuming this ergodicity we obtain numerically the following results for the statistics of knotted polygons in the limit of a low crossing fugacity we find a localization along the polygon of all the primary factors forming the knot increasing the crossing fugacity gives rise to a transition from a self avoiding walk to a branched polymer behaviour

the green kubo formula and power spectrum of reversible markov processes as is known the entropy production rate of a stationary markov process vanishes if and only if the process is reversible in this paper we discuss the reversibility of a stationary markov process from a functional analysis point of view it is shown that the process is reversible if and only if it has a symmetric markov semigroup equivalently a self adjoint infinitesimal generator applying this fact we prove that the green kubo formula holds for reversible markov processes by demonstrating that the power spectrum of each reversible markov process is lorentz typed we show that it is impossible for stochastic resonance to occur in systems with zero entropy production

stretched polygons in a lattice tube we examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f the existence of the limiting free energy for these so called stretched polygons is proved and then using transfer matrix arguments a pattern theorem for stretched polygons is proved note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f while without the tube constraint it has so far only been proved for large values of f the stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n edge stretched polygon approaches a positive value non decreasing in f as n we then show that the knotting probability of an n edge stretched polygon confined to a tube goes to one exponentially as n thus as n when polygons are influenced by a force f no matter its strength or direction topological entanglements as defined by knotting occur with high probability

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

knotting probabilities after a local strand passage in unknotted self avoiding polygons we investigate both theoretically and numerically the knotting probabilities after a local strand passage is performed in an unknotted self avoiding polygon sap on the simple cubic lattice in the polygons studied it is assumed that two polygon segments have already been brought close together for the purpose of performing a strand passage this restricts the polygons considered to those that contain a specific pattern called at a fixed location an unknotted polygon containing is called a sap it is proved that the number of n edge saps grows exponentially with n at the same rate as the total number of n edge unknotted saps those with no prespecified strand passage structure furthermore it is proved that the same holds for subsets of n edge saps that yield a specific after strand passage knot type thus the probability of a given after strand passage knot type does not grow or decay exponentially with n instead it is conjectured that these after strand passage knot probabilities approach as n goes to infinity knot type dependent amplitude ratios lying strictly between 0 and 1 this conjecture is supported by numerical evidence from monte carlo data generated using a composite aka multiple markov chain monte carlo bfacf algorithm developed to study saps a new maximum likelihood method is used to estimate the critical exponents relevant to this conjecture we also obtain strong numerical evidence that the after strand passage knotting probability depends on the local structure around the strand passage site if the local structure and the crossing sign at the strand passage site are considered then we observe that the more compact the local structure the less likely the after strand passage polygon is to be knotted this trend for compactness versus knotting probability is consistent with results obtained for other strand passage models however we are the first to note the influence of the crossing sign information we use two measures of compactness one involves the size of a smallest polygon that contains the structure and the other is in terms of an opening angle the opening angle definition is consistent with one that is measurable from single molecule dna experiments the theoretical and numerical approaches presented here are more broadly applicable to other sap models

the expected total curvature of random polygons we consider the expected value for the total curvature of a random closed polygon numerical experiments have suggested that as the number of edges becomes large the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant we show that this is true for a natural class of probability measures on polygons and give a formula for the constant in terms of the moments of the edgelength distribution we then consider the symmetric measure on closed polygons of fixed total length constructed by cantarella deguchi and shonkwiler for this measure we are able to prove that the expected value of total curvature for a closed n gon is exactly pi over 2 n pi over 4 2n over 2n 3 as a consequence we show that at least 1 3 of fixed length hexagons and 1 11 of fixed length heptagons in bbb r 3 are unknotted

the statistics of collapsing square lattice trails with a fixed number of vertices of degree 4 a trail on the square lattice with a fixed number k of vertices of degree 4 is called a k trail we model polymer collapse using k trails by incorporating an interaction energy which is proportional to the number of nearest neighbour contact edges of the trail it is known that the number of square lattice n edge closed open k trails can be bounded above and below to o nk by the number of n step self avoiding circuits walks this along with pattern theorems for self interacting self avoiding circuits and walks are used herein to establish upper and lower bounds to o nk for the collapsing free energy of k trails in terms of self avoiding circuits or walks as appropriate we also use pattern theorems to obtain bounds on the limiting nearest neighbour contact density for collapsing k trails finally we investigate k trails with a fixed density of nearest neighbour contacts and show that their limiting entropy per monomer is independent of k

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

computing the writhe on lattices given a polygonal closed curve on a lattice or space group we describe a method for computing the writhe of the curve as the average of weighted projected writhing numbers of the polygon in a few directions these directions are determined by the lattice geometry the weights are determined by areas of regions on the unit 2 sphere and the regions are formed by the tangent indicatrix to the polygonal curve we give a new formula for the writhe of polygons on the face centred cubic lattice and prove that the writhe of polygons on the body centred cubic lattice the hexagonal simple lattice and the diamond space group is always a rational number and discuss applications to ring polymers

knotted polygons with curvature in the knot probability of semiflexible polygons on the cubic lattice is investigated the degree of stiffness of the polygon is mimicked by introducing a bending fugacity conjugate to the curvature of the polygon by generalizing kesten s pattern theorem to semiflexible walks we show that for any finite value of the bending fugacity all except exponentially few sufficiently long polygons are knotted

the green kubo formula autocorrelation function and fluctuation spectrum for finite markov chains with continuous time a general form of the green kubo formula which describes the fluctuations pertaining to all the steady states whether equilibrium or non equilibrium for a system driven by a finite markov chain with continuous time briefly mc is shown the equivalence of different forms of the green kubo formula is exploited we also look at the differences in terms of the autocorrelation function and the fluctuation spectrum between the equilibrium state and the non equilibrium steady state also if the mc is in the non equilibrium steady state we can always find a complex function such that the fluctuation spectrum of is non monotonous in 0

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

on the dominance of trivial knots among saps on a cubic lattice the knotting probability is defined by the probability with which an n step self avoiding polygon sap with a fixed type of knot appears in the configuration space we evaluate these probabilities for some knot types on a simple cubic lattice for the trivial knot we find that the knotting probability decays much slower for the sap on the cubic lattice than for continuum models of the sap as a function of n in particular the characteristic length of the trivial knot that corresponds to a half life of the knotting probability is estimated to be 2 5 105 on the cubic lattice

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

on characterization of reversible markov processes by monotonicity of the fluctuation spectral density for the system driven by a stationary markov process t t r if it is in nonequilibrium steady state i e irreversible then there exists a function such that the fluctuation spectrum or say power spectrum density of t is nonmonotonic in 0 under mild conditions which means that there exist nonzero spectrum peaks of the fluctuation spectrum for the system driven by a markov chain with discrete time t t z even if it is in equilibrium state i e reversible one cannot distinguish the equilibrium and nonequilibrium steady state in terms of the monotonicity of the fluctuation spectrum any more

the mean squared writhe of alternating random knot diagrams the writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram it plays an important role not only in knot theory itself but also in various applications of knot theory to fields such as molecular biology and polymer physics the mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one half however such a diagram is usually not minimal if we restrict ourselves to a minimal knot diagram then the choice of which strand is the over or under strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams this paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced alternating knot diagrams we demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates in particular this is true when the sample space contains only diagrams of a special structure when the sample space is large enough to contain not only diagrams of a special type then the mean squared writhe for n crossing diagrams tends to grow linearly with n but at a faster rate than n indicating an intrinsic property of alternating knot diagrams studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used which is an important area of study in the applications of random knot theory