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of interest to anglers seeking to fill their creels and children seeking to fasten their shoes , a wide audience has found knots compelling from time immemorial . in the scientific community , knots have been featured in initial formulations of the nature of atoms , @xcite ( see a popular historical account in @xcite ) , the formulation of certain path integrals , @xcite , and also in quantitative biology , where knots have been observed in , @xcite , and tied into , dna , @xcite , where the space of knots is biologically created and manipulated @xcite . knots also have been observed occasionally in proteins , @xcite . historically , the classification of knots and study of knot invariants were the first subjects of knot theory @xcite , and this remains in the center of attention among knot theorists of mathematical orientation @xcite . another fundamental aspect of knot theory is that of knot entropy . physically , this group of problems comes to the fore in the context of polymers and biophysics . mathematically , this issue belongs to both topology and probability theory and seems to remain underappreciated in the mathematics and mathematical physics community . even the simplest question in this area is poorly understood : what is the probability that a randomly closed loop in @xmath5 will be topologically equivalent to plane circle ? in other words , using professional parlance of the field , what is the probability that random loop is a trivial knot ( unknot ) , @xmath6 ? there are , of course , many more questions along the same lines , e.g. , what are probabilities of other more complex knots ? what is the entropic response of a topologically constrained loop to various perturbations , etc . most of what we know about these `` probabilistic topology '' questions is learned from computer simulations . in particular , it has been observed by many authors over the last 3 decades @xcite that the trivial knot probability depends on the length of the loop , decaying exponentially with the number of segments in the loop , @xmath0 : @xmath7 for some lattice models this exponential law , in the @xmath8 asymptotics , was also mathematically proven @xcite . it was also noticed @xcite that the same exponential law , with the same decay parameter @xmath2 , also describes the large @xmath0 asymptotical tail of the abundance of any other particular knot - although for complex knots exponential decay starts only at sufficiently large @xmath0 ( as soon as the given knot can be identified as an underknot @xcite ) . an alternative view of formula ( [ eq : triv_p ] ) , useful in the context of thermodynamics , implies that the removal of all knots from the loop is associated with thermodynamically additive ( linear in @xmath0 ) entropy loss of @xmath9 per segment ; in other words , at the temperature @xmath10 , untying all knots would require mechanical work of at least @xmath11 per segment . another manifestation of the importance of the @xmath2 parameter was found in the recent series of works @xcite . these works belong to the direction @xcite addressing the spatial statistics of polymer loops restricted to remain in a certain topological knot state . it turns out that even for loops with no excluded volume and thus are not self - avoiding , @xmath2 marks the crossover scale between mostly gaussian ( @xmath12 ) and significantly non - gaussian ( @xmath13 ) statistics . indeed , at @xmath12 , locking the loop in the state of an unknot excludes only a small domain of the conformational space which produces only marginal ( albeit non - trivial @xcite ) corrections to gaussian statistics - for instance , mean - squared gyration radius of the loop is nearly linear in @xmath0 . by contrast , at @xmath13 , the topological constraints are of paramount importance , making the loop statistics very much non - gaussian , and consistent with effective self - avoidance @xcite . thus , it seems likely that the parameter @xmath2 might hold the key to the entire problem of knot entropy . we therefore decided to look at this parameter more closely in this paper . present understanding of the values of @xmath2 is quite modest . first , the constant s value was invariably found to be quite large , around @xmath14 for all examined models of `` thin '' loops with no excluded volume , or no self - avoidance @xcite . second , it is known that knots are dramatically suppressed for `` thick '' self - avoiding polymers , which means that @xmath2 rapidly increases with the radius of self - avoidance @xcite . the latter issue is also closely connected to the probabilities of knots in lattice models , where the non - zero effective self - avoidance parameter is automatically set by the lattice geometry . in the present paper , we will only consider the arguably more fundamental case of loops with no self - avoidance . the starting point of our analysis is the observation that @xmath2 appears to be noticeably different for two standard polymer models for which common sense suggests that they should be equivalent . both models can be called freely - jointed in the sense that they consist of @xmath0 rigid segments with free rotation in the joints . however , in one model all segment vectors are of the same length , while in the other model segment vectors are taken from a gaussian distribution . the motivation to consider the gaussian distributed step vectors comes from the idea of decimation , or renormalization : we can start from the loop with @xmath15 segments of equal length and then group them into @xmath16 blobs of @xmath17 bare segments , each blob having nearly gaussian distributed end - to - end vector . with respect to the knot abundance , the fixed length model was examined in @xcite and the gaussian model in @xcite . it was noticed that @xmath2 for the gaussian distributed steps was larger than for identical steps , assuming no self - exclusion in both cases . no attention was paid to this observation , possibly because there was no confidence that the observed difference is real , in the context of the numerical error bars in the pertinent measurements . recently , @xcite , more detailed data became available which suggest that indeed @xmath2 is different for the two models , with fixed or gaussian distributed steplength . a similar result was independently obtained by vologodskii @xcite . latter in this article , we present even better quality data supporting the same observation that @xmath2 is different for these two models . this is a rather disturbing observation . indeed , the idea of universality in polymer physics @xcite suggests that there should not be any difference between these two models as far as any macroscopic quantity is concerned . for instance , not only is the mean squared gyration radius the same for both models , but even the distribution of the gyration radii are the same , except far into the tails . in general , the difference between polymer models of this type becomes significant only in the strong stretching regime @xcite or at high density @xcite . even if one takes into account the idea that knots , when present , are most likely localized along the chain @xcite , it is unclear how this fact can manifest itself for the loop that has no knots . thinking generally about the loop models with fixed or gaussian steplength , our reaction to this discrepancy is to realize that the major difference between the two freely - jointed loop models is that the gaussian model may have a few unusually long segments , suppressing the ability of other shorter segments to wind around , and thus decreasing the possibility for knots to occur , and accordingly , increasing @xmath2 . thus , the ability to take long strides might account for the comparative slowness of the ensemble of loops with gaussian steplengths to diversify their knot spectrum with increasing @xmath0 . the main goal of the present work is to investigate this conjecture . the plan of the work is as follows . after a brief description of our computational algorithms used to generate closed loops and to identify their topologies ( section [ sec : methods ] ) , we present computational results ( section [ sec : results ] ) on knot abundance for a variety of models differing in the width of their steplength distribution . in addition to the already mentioned loops with fixed , and gaussian - distributed steplengths , in order to look at the even broader distributions which allow for very long segments , we also generated loops with the generalized cauchy - lorentz `` random - flight '' distribution . finally , we include loops of bimodally distributed fixed steplength .    in brief , our results are as follows . first , we confirm the exponential decay law of the unknot probability , formula ( [ eq : triv_p ] ) , across all models examined . second , we find qualitatively that indeed a wider distribution of the segment lengths leads to knot suppression , ie a larger @xmath2 . third , and most unexpectedly , we find that @xmath2 does not show any signs of any singular behavior associated with the divergence of mean squared segment length or any other moment of the segment length distribution . instead , @xmath2 blows up and appears to grow without a bound when the distribution of segment lengths approaches the border of normalizability . all polymer models referenced and employed in this work use a freely jointed model to represent a polymer loop . the polymer is represented by a set of @xmath0 vertices in 3d , with position vectors @xmath18 , where the step between successive vertices is described , @xmath19 . in all models , we assume that the distribution of segment vectors @xmath20 , which we call @xmath21 , is spherically symmetric and depends only on the steplength @xmath22 , such that @xmath23 we also assume that the mean squared steplength is always the same ( when defined ! - see below ) , we denote it @xmath24 : @xmath25 with this in mind , the simplest measure of the distribution breadth involves higher order moments : @xmath26 where @xmath27 . specifically , we analyzed the following models . the * fixed steplength model * is described by the distribution @xmath28 for this model , of course @xmath29 . with @xmath0 segments , the loop s contour length is obviously @xmath30 and the mean squared gyration radius of the loop is @xmath31 . the * gaussian steplength model * is generated by the distribution @xmath32 \ , \ ] ] in this case , @xmath33 . the contour length of the @xmath0-segment loop in this model is @xmath34 and the mean squared gyration radius is @xmath35 . the * random flight steplength model * is obtained from the generalized cauchy - lorenz distribution ( also known in the theory of lvy flights @xcite ) of the form @xmath36 \right/ \left ( 4 \ell^3 \pi^2 \right ) } { 1+(r/\ell)^{\alpha } } \ , \label{eq : mcl_prob}\ ] ] where the factor in the numerator ensures normalization . here , @xmath37 , a parameter of the distribution , must be greater than @xmath38 , otherwise the normalization integral diverges . nevertheless , it would be fair to speak about a family of random - flight models , parameterized by @xmath39 , instead of just one model . varying @xmath37 allows us to work with a `` tunable '' distribution . these distributions `` fat '' power - law tails lead to diverging moments ( which is why they are used to describe super - diffusive behavior seen in biological foraging @xcite , and quite recently , in the diffusion of bank - notes across the united states , @xcite ) . specifically , the ensemble - averaged contour length of the loop is well defined only at @xmath40 ( @xmath41 ) , mean squared gyration radius exists at @xmath42 , and @xmath43 only exists at @xmath44 , in which case it is equal to @xmath45 ^ 2 } { \sin\left [ 3 \pi / \alpha \right ] \sin\left [ 7\pi / \alpha \right]}-1 } \ \ \ ( { \rm at } \ \alpha > 7 ) \ .\ ] ] finally , we also include loops with * bimodally distributed steplength*. for these loops , @xmath46 which means that two possible steplengths , @xmath47 or @xmath48 , occur with probabilities @xmath49 and @xmath50 , subject to the normalization conditions , @xmath51 , and @xmath52 . all bimodally distributed models can be conveniently parameterized by @xmath49 and @xmath53 . for these models , @xmath54 might be very large if @xmath55 is very large and @xmath49 is rather close to unity ( @xmath56 ) . unbiased generation of closed loops is of decisive importance for our work . recently , we gave a detailed review of the existing computational methods to generate statistically representative closed loops ( see the last section of the work @xcite ) . in principle , the best way to generate loops is based on the so - called conditional probability method . the idea is that a closed path is generated as a random walk , step by step , except after the completion of @xmath57 steps , the next step , @xmath58 , is generated from the analytically computed probability distribution of the step vector @xmath20 , subject to the condition that after @xmath59 more steps , the walker returns to the starting point . this idea was first suggested and implemented for gaussian distributed steps @xcite . recently , we implemented this method for steps of equal length @xcite . unfortunately , this method is computationally costly , and appears to be prohibitively difficult to implement for more sophisticated models , such as random flight . we therefore use the simpler method , called the method of triangles . this method @xcite generates loops of @xmath0 segments with @xmath0 divisible by @xmath60 . it involves creating a set of @xmath61 equilateral triangles , each randomly oriented in 3d space . each triangle is considered a triplet of vectors with zero sum . a random permutation of the @xmath0 edge vectors which make up these @xmath61 triangles , and then connecting all @xmath0 vectors head - to - tail , creates a loop which will be closed , as the @xmath0 bond vectors together have @xmath62 vector sum . of course , this method imposes correlations between segments . we therefore take special care to compare the results of this method with the unbiased generation using the conditional probability method for both gaussian distributed and equal length step models . we found that no appreciable deviations in knot abundance data arise from the imperfection of the triangle method . we therefore use the method of triangles to generate the random flight and bimodal distributed loops , for which no alternative method is available . to avoid even the slightest problems with correlations , implicit in our triplet method , and to ensure that the decay of trivial knot probability is in the exponential regime , we exclude the data from small loops and fit the trivial knot probability on the interval @xmath63 $ ] .      for each of the models listed above , we generated loops of up to at least @xmath64 segments . once the loop was generated , its knotted state was assessed computationally with the alexander determinant , @xmath65 , as well as the vassilev invariants of degree @xmath66 and @xmath60 , @xmath67 and @xmath68 . for this purpose , we employed knot analyzing routines described in detail elsewhere @xcite . the fraction of generated loops with trivial topology was recorded for each loop type . in the interval @xmath69 , every sample consisted of at least @xmath70 loops . as a result , the plot of trivial knot probability was created for each model , with statistical error bars smaller than the data points in figure [ fig : compare_fixed_and_gaussian ] . based on the data , @xmath2 was measured for every model in our repertoire . our main results are shown in figure [ fig : compare_fixed_and_gaussian ] . there , we present the semi - log plots of the data on the trivial knot probability as a function of the number of segments @xmath0 for a variety of models . to begin with , all our data agree quite well with the exponential character of the @xmath71 dependence , formula ( [ eq : triv_p ] ) . our main emphasis is therefore the study of the characteristic value of @xmath2 for various models . let us start with the two most commonly used models . in the case of fixed length steps , we obtain @xmath72 ( see also @xcite ) . by contrast , for gaussian distributed length steps we get @xmath73 ( see @xcite ) . all data for the gaussian model was generated using the fundamentally unbiased conditional probability method . for the model with fixed length steps , we compare in figure [ fig : checking_triangles ] the data obtained by the conditional probability method @xcite and data generated by the much more efficient method of triangles ( see above section [ sec : generation ] ) . as the figure indicates , there is practically no visible difference in the results . accordingly , we unreservedly rely on the triangles method in the rest of this work . although our main attention in this paper is on the trivial knots ( unknots ) , we show in figure [ fig : other_knots ] some data for more complex knot probabilities . our data at least do not contradict the assertion that the probability of every particular knot decays exponentially at sufficiently large @xmath0 , with the same decay length as the trivial knot probability . at the same time , our data also confirm the systematic difference between models - decay length , although the same for all types of knots , does depend on the distribution of step lengths involved . specifically , probabilities of knots @xmath74 and @xmath75 decay at a noticeably slower rate for the gaussian model than for the model with fixed steps .      the results of our study of loops with random - flight steplength , @xmath76 $ ] , are summarized in figures [ fig : compare_fixed_and_gaussian ] and [ fig : mcl_fit ] . the raw curves of probability given in figure [ fig : compare_fixed_and_gaussian ] clearly show that the odds of finding an unknot in a set of loops get increasingly unfavorable as @xmath37 decreases . that is not unexpected : at smaller @xmath37 , the probability distribution ( [ eq : mcl_prob ] ) acquires an increasingly fat tail , which implies the presence of a fraction of exceptionally long segments , and they of course suppress the chance of knots .    at very large @xmath37 , the knot probability for the random flight model appears similar to the data for fixed steplength loops . indeed , as figure [ fig : mcl_fit ] indicates , @xmath2 for the random flight model at very large @xmath37 approaches @xmath77 which is not dramatically different from @xmath72 for the fixed length steps . in fact , the remaining difference might be associated with the fact that even at very large @xmath37 , the random flight model , although it has essentially no very long steps , has some relatively short ones , which might account for the discrepancy in @xmath2 . figures [ fig : compare_fixed_and_gaussian ] and [ fig : mcl_fit ] show further that at @xmath78 , the random - flight loops behave essentially the same way as gaussian steps in terms of @xmath2 . most interestingly , figure [ fig : mcl_fit ] shows no sign of anything unusual happening to @xmath2 at the values of @xmath37 at which various physically important moments of the segment length distribution ( [ eq : mcl_prob ] ) start diverging . for instance , at @xmath79 the mean squared gyration radius diverges , at @xmath80 even the contour length of the loop diverges - and yet none of these facts find any visible reflection on the dependence of @xmath2 on @xmath37 . @xmath2 keeps smoothly increasing with @xmath37 , with a maximum measured value of @xmath81 at @xmath82 . it appears that @xmath2 in fact blows up and goes to infinity as @xmath83 approaches @xmath60 - the border below which the distribution ( [ eq : mcl_prob ] ) is not normalizeable . moreover , as the inset of figure [ fig : mcl_fit ] shows , this divergence is well approximated by the power law dependence of the form , @xmath84 in fitting the data with this power law , we ignored the small irregularities visible around @xmath85 ( or @xmath86 \approx 2.2 $ ] ) , which we attribute to the numerical problems with the cutoff implemented in our use of eq . ( [ eq : int_prob ] ) . the meaning and physical origin of the apparent criticality observed at @xmath37 approaching its minimal possible value of @xmath60 currently evades our understanding .      for this model , we examined the interval @xmath87 $ ] , with at least @xmath70 loops in each simulation record for @xmath88 , and at least @xmath89 loops in those records used with length @xmath90 . we also consider parameters @xmath49 and @xmath91 in the intervals @xmath92 $ ] and @xmath93 $ ] , respectively . by symmetry , it is sufficient to look at @xmath94 ; this means , @xmath49 is the fraction of _ shorter _ segments . the raw data from the loops with bimodal steplength are presented in figure [ fig : bimodal_surface ] . this surface plot charts the change in @xmath2 as a function of two parameters of the model , @xmath49 and @xmath55 . below the surface is the corresponding contour plot of @xmath2 as a function of these two parameters . the changes in @xmath2 are smooth , and there do not seem to be any singularities in the behavior of @xmath2 . the maximum observed value of @xmath95 occurs at @xmath96 , @xmath97 . this maximum of @xmath2 appears to be rather sharp , small deviations in @xmath49 from @xmath98 lead to smaller @xmath2 values . from the data , we believe that in this model @xmath2 is maximized when the fraction of shorter segments is @xmath99 . as regards the second degree of freedom , @xmath55 , while our data certainly shows that as @xmath100 the bimodal system approaches the fixed steplength system of section [ fixed_step_loops ] ( ie @xmath101 ) , in the opposite direction of @xmath102 , it is not clear if @xmath2 will continue to increase in an unbounded way or be in some way encumbered . in this regard our present knot analysis machinery is limited by the relative disparity between segments of different length , and more work needs to be done to elucidate the scaling of @xmath2 in the limit @xmath103 . qualitatively , all of our data are consistent with the idea that what suppresses knots is the presence of a fraction of unusually long segments . one could then hypothesize that @xmath2 might depend on some unique property of the segment length distribution , for instance , @xmath43 , as defined in eq . ( [ eq : sigm_def ] ) . this hypothesis is tested in figure [ fig : bimodal_compare_small ] ; the figure indicates that the hypothesis fails . nevertheless , the results presented in this figure are interesting , as they show that large @xmath2 can be achieved by the combination of segment length difference ( @xmath55 ) and segment type fractions . although a functional relationship between @xmath2 and @xmath37 seems evident in our data for random flight loops , the data we have for loops with bimodal distribution of steplength do not suggest a simple single parameter which determines @xmath2 . in a qualitative sense , it does seem that there exists a relationship between @xmath2 and the reach of successive segments within the chain ( as seen in figure [ fig : bimodal_compare_small ] ) . it seems qualitatively clear indeed that knottedness is greatly suppressed by the presence of some very long segments . thus , the slowly decaying tail of the segment length probability distribution , or the presence of a small fraction of very long segments , implies a large @xmath2 , which sounds natural . however , our understanding of these observations beyond the qualitative level is limited . we found that @xmath2 appears to exhibit no singularity associated with divergence of any natural characteristics of the loop , such as its gyration radius or even contour length ; instead , @xmath2 exhibits power law critical behavior when the segment length distribution approaches the boundary of normalizability . in addition , we do not know which property of the segment length distribution determines @xmath2 . we have established that this is not simply the distribution width , and from the results with random flight distribution it seems also clear that it is not based on any finite moments of the distribution . we consider the development of a fuller understanding of the variance of @xmath2 a compelling challenge . the data clearly show that wide variation in the behavior of @xmath2 is possible , including very large values of @xmath2 in certain models . given that @xmath2 plays the role of the cross - over length for the critical behavior of topologically constrained loops @xcite , we can speculate that the models with very large @xmath2 are in some way similar to the models of self - avoidance in the vicinity of the @xmath104-point . we think that this analogy deserves very close attention . we acknowledge useful discussion with a.l . efros and a.  vologodskii . we express thanks to r. lua for the use of his knot analysis routines . we are grateful for access to the minnesota supercomputing institute s ibm core resources and the university of minnesota physics department linux cluster , resources which made much of the simulation possible . this work was supported in part by the mrsec program of the national science foundation under award number dmr-0212302 . to generate steplengths from the random - flight distribution ( [ eq : mcl_prob ] ) , we first take a random number @xmath105 from the uniform distribution on the interval @xmath106 $ ] and then find steplength @xmath107 as @xmath108 , where the mapping @xmath109 is determined by the equation @xmath110 where @xmath111 is given by eq . ( [ eq : mcl_prob ] ) . although a closed - form representation of the right - hand side of eq . 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'' from mathworld  a wolfram web resource . ` http://mathworld.wolfram.com/levydistribution.html `
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 @xmath0 segments follows a decaying exponential form , @xmath1 , where @xmath2 marks the crossover from a mostly unknotted ( ie topologically simple ) to a mostly knotted ( ie topologically complex ) ensemble . in the present work we use computational simulation to look closer into the variation of @xmath2 for a variety of polymer models . among models examined , 
 @xmath2 is smallest ( about @xmath3 ) for the model with all segments of the same length , it is somewhat larger ( @xmath4 ) for gaussian distributed segments , and can be very large ( up to many thousands ) when the segment length distribution has a fat power law tail .