The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. Here, we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages. In addition, we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot. These numbers include: crossing number; average three-dimensional writhe; number of topological domains; and the average relaxation value.
During the 1860s, British physicist William Thomson (now remembered as Lord Kelvin) proposed that different types of atoms were simply different types of knots made of vortices of ether (a hypothetical substance believed to be necessary for the propagation of electromagnetic radiation; Thomson 1869). As we now know, further development of physics negated the existence of ether and provided another explanation of distinct properties of various atoms. However, the hypothesis of Kelvin needs to be seen as a proposal that real elementary particles (which Thomson thought were atoms) can have a form of knots. This latter interpretation has recently found a spectacular confirmation (Buniy & Kephart 2003; Buniy & Kephart 2005). A class of elementary particles known as glueballs have such a spectrum of mass that clearly indicates that these particles exist as knotted quantum chromodynamics flux lines (Ralston 2003), where the flux lines follow paths of ideal knots, i.e. the shortest possible cylindrical tubes that can be closed into a given knot type (Moffatt 1990; Katritch et al. 1996; Faddeev & Niemi 1997). Inspired by this recent discovery, we present a natural classification of knots (with up to 9 crossings) that reflects the relationships between various knots and that takes into account earlier observations that the writhe, which is a measure of chirality of closed curves in a three-dimensional space, is additive and quantized in the case of ideal geometric representations of knots (Katritch et al. 1997; Pieranski 1998; Cerf & Stasiak 2000; Devlin 2001; Pieranski & Przybyl 2001; Cerf & Stasiak 2003).
To obtain a random configuration of a given knot type, we start with a polygonal configuration that resembles the axial trajectory of the so-called ideal knots of a given type (Katritch et al. 1996). These starting configurations are then evolved as non-phantom chains by applying random crankshaft moves (Vologodskii et al. 1992). The non-phantom evolution is achieved by accepting only these crankshaft moves, during which there are no segment–segment passages. After 20 000 of such accepted random moves, when the configuration is sufficiently randomized, the polygon is further evolved by random crankshaft rotations, but, in addition, moves that result in just one intersegmental passage are also accepted. The occurrence of intersegmental passages was detected by the checking for intersections between the surface of revolution generated by the rotating portion of the chain and segments of the subchain that was not rotated. After the first move that resulted in just one intersegmental passage, the evolution is terminated and the knot type of the polygonal chain is determined by the calculation of its HOMFLY polynomial (Freyd et al. 1985; Ewing & Millett 1996; Dobay et al. 2003). The entire procedure is repeated 20 000 times for each knot type with up to 9 crossings in the minimal crossing representation.
(a) The evolutionary tree of knots
We classify knot types according to their reciprocal kinship in the knot space. For knot types that are directly related to each other, we consider the knot types that can be converted into each other by one intersegmental passage (figure 1c). Using a biological example, we consider a pair of knot types as related to each other when a DNA knot of a given type can be converted into a knot of another type by one catalytic cycle of a DNA topoisomerase (Darcy & Sumners 2000; Flammini et al. 2004). We build a genealogical, or rather an evolutionary, tree of knots by investigating the inter-knot transitions resulting from random intersegmental passages, occurring in random configurations of each of the 101 types of knots that have up to 9 crossings in their standard diagrams with minimal numbers of crossings. The detailed procedure is described in §2. In brief, 20 000 random configurations of a given knot constructed out of 32 segment-long equilateral polygons are permitted to undergo random crankshaft motion, and after the first intersegmental passage, the evolution is terminated and the knot type of the polygonal chain is characterized by the calculation of its HOMFLY polynomial (Freyd et al. 1985; Ewing & Millett 1996; Dobay et al. 2003). For the construction of the tree of knots, we take into account only one-passage connectivity between the knots that differ in their minimal number of crossings, where, in each pair of such knots, the simpler one (with smaller minimal number of crossings) is considered as a predecessor and the more complex one as a successor (figure 1c). Most often successors have two more crossings than predecessors have. Therefore, we concentrate first on the predecessors and the successors that differ in their minimal number of crossing by 2. A given knot type frequently has more than one predecessor with two fewer crossings. In such a case, we measure the relative order of kinship between the successors and the predecessors by determining the probability of relaxation of individual knot types towards their respective predecessors. We include this information in the constructed tree of knots (figure 2). In some knot types, random intersegmental passages lead with nearly identical probability (relative differences smaller than 5%) to two or more different predecessors. In such a case, we assign the same degree of kinship between such a successor knot and each of its respective predecessors. When a given knot type does not have a predecessor with two fewer crossings, we look for its predecessors with a maximal possible number of crossings and indicate this accordingly on the evolutionary tree of knots.
It was observed earlier that the three-dimensional writhe, which is a measure of chirality of closed curves in space, is additive and quantized for axial trajectories of ideal geometric representations of knots and that this applies also to the mean writhe of statistical ensembles of random knots of a given type (Katritch et al. 1997; Pieranski 1998; Cerf & Stasiak 2000; Pieranski & Przybyl 2001). The additivity of writhe implies that writhe values contributed by different types of crossings simply sum up to the final writhe of the entire knot. Quantization of writhe is based on the observation that crossings constituting a knot or a link introduce just four quantized values of writhe +4/7, −4/7, +10/7 and −10/7 that depend on the character (parallel or antiparallel) and the sign of the crossing (Stasiak 2000; see figure 1 for the explanations concerning the signs and characters of crossings). Therefore, the writhe value of a given knot provides the information about the types of crossings that constitute a given knot. For this reason, we decided to arrange the knots on their evolutionary tree according to their number of crossings (ordinate value) and their writhe value (abscissa).
Figure 2 presents the evolutionary tree for knots with up to 9 crossings. To allow us better insight into the structure of the tree, we have ‘split’ it into two weakly interconnected parts: the first leading to knots with 6 and 8 crossings (lower part) and the second presenting positions of knots with 5, 7 and 9 crossings (upper part). To avoid repetitions, we have presented only the central and right half of the tree that together include achiral knots and right-handed enantioforms of chiral knots with up to 9 crossings. The left half of the tree with left-handed enantioforms is simply the mirror image of the right half; therefore, it is not shown, with the exception of left-handed enantioforms of the knots 61 and 77. The branches of the tree drawn as red lines in figure 2 connect these knots that can be converted into each other by intersegmental passages which remove (going down the branch) or introduce (going up the branch) parallel crossings into the minimal diagram of a given knot (see figures 1 and 3 for the explanations of parallel (P) and antiparallel (A) crossings). The branches drawn in blue correspond to these intersegmental passages that remove or introduce antiparallel crossings into a minimal diagram of a given knot. Branches leading from left down to right up correspond to intersegmental passages that introduce crossings of positive sign, while branches inclined toward the left indicate intersegmental passages introducing negative crossings. Note, though, that a drawn branch connecting two groups of knots indicates that only some knots in these two groups can be converted into each other by one intersegmental passage of a given type. To find out which knots form predecessor–successor pairs, one needs to use the colour code applied in figure 2. To explain it, let us take an example of the 99 knot and its predecessors (figure 3). Analysing random intersegmental passages occurring in random configurations of 99 knots composed of 32 freely jointed segments, we observed that out of more than 14 000 such passages leading to the creation of simpler knots, 42.5% resulted in a conversion into the 73 knot. Conversion into the 75 and 71 knots were observed in 35.7 and 21.8% of the corresponding cases, respectively. To reflect these results in figure 2, the knot 99 (in the penultimate group of 9 crossing knots) is marked with three coloured squares of different sizes. The biggest square is yellow, indicating that the 99 knot most frequently relaxes to the 75 knot (placed on a yellow background field in figure 2). The second biggest square has the same colour as the background field of 73 and the third square in respect of size has the colour of the background field of the 71 knot. Note that the colour code is applied to all knots with seven or more crossings that have predecessors with two fewer crossings and serves two functions: it informs which of the knots with two fewer crossings are indeed the predecessors of a given knot and it provides information about the order of relatedness of a given knot with these predecessors in case there are more of them. Achiral knots (these with zero writhe) undergo random passages to right- and left-handed predecessors with equal probability, but only right-handed predecessors are presented in the figure. Several knots do not have the predecessors with two fewer crossings. In such a case, we indicate what their most complex predecessors are. Thus, knots 928 and 940 have knot 821 as their predecessor, and this is also indicated with the colour code. Knots 818 and 931 have their predecessors (knots 31 and 63, respectively) marked with the corresponding letters ‘a’ and ‘b’. In figure 2, we did not include the actual conversion rate to different predecessors obtained in the simulation, but reported only the relative order of relatedness to the respective predecessors observed in the simulations. In this respect, it is interesting to mention the case of the knot 86 for which we analysed over 14 000 passages leading to a simplification in knots topology, and out of those 34.8% led to a passage into the 61 knot and the same percentage led to the 62 knot. Therefore, in figure 2, the 86 knot is marked with two differently coloured squares of the same size.
(b) Minimal diagrams of alternating knots reveal their relatedness with all predecessors
Returning to the example of the 99 knot, it is important to mention that out of 20 000 analysed random passages, all those that led to a conversion to simpler knots (with smaller number of crossings than 9) resulted either in the creation of 71, 73 or 75 knots. Figure 3 shows that the minimal diagram of the knot 99 allows us to predict not only all the possible predecessors of this knot, but also the relative kinship of this knot to its predecessors. If one takes nine minimal diagrams of knot 99 and performs a strand passage at a different crossing in each diagram, then, in four out of nine cases (44.5%), the resulting knot will be 75, in three cases (33.3%) the 73 knot is created and the 71 knot is created in two cases (22.2%). Note a close correspondence between the conversion rates to three different predecessor knots resulting from simulated random passages in random configurations of the 99 knot (see §3a), and these resulting from a simple analysis of the standard, minimal diagram of the 99 knot. In fact, in all the analysed alternating knots, we observed that their minimal diagrams allowed us not only to predict all their predecessor knots (not only those with two fewer crossings), but also to closely estimate the rate of random passages into all predecessor knots of a given successor knot. As could be expected, based on the analysis of random passages, the minimal diagram of the 86 knot has an equal number of crossings that lead to a passage into the 61 and 62 knots. We conjecture that for sufficiently long chains, the proportions between different predecessor knots obtained by random passages will correspond to corresponding proportions that can be predicted based on the minimal diagram of a given alternating knot.
Recall that the three-dimensional writhe value of statistical ensembles of random knots of a given type can be accurately predicted just from the minimal diagrams of alternating knots (Cerf & Stasiak 2000). Our observation concerning random passages provides another example that minimal diagrams of alternating knots capture the important information about the statistical behaviour of the corresponding random knots.
(c) Non-alternating knots
The branches of the tree of knots drawn as green lines in figure 2 correspond to intersegmental passages leading to the creation or relaxation of non-alternating knots. The writhe differences between ideal geometric forms of alternating knots with n crossings and their non-alternating successor knots with n+2 crossings are about 2 or −2, depending on the sign of introduced crossings (Pieranski 1998). This difference of writhe corresponds therefore to the sum of contributions of parallel and antiparallel crossings of the same sign (10/7 and 4/7; see inset in figure 2). Consequently, to calculate the expected writhe of non-alternating knot from its minimal diagram, we associate a change of writhe of 2 or −2 to the passage that eliminates two crossings (of positive or negative sign) and leads to the minimal crossing diagram of an alternating knot with two fewer crossings.
(d) Topological domains
While the disposition of knots according to their minimal crossing number and writhe allowed us to arrange knots into different groups and conveniently present their relatedness, it is necessary to distinguish the knots within groups that have the same writhe and the same crossing number. To this aim, we ‘decompose’ the knots not only into their elementary elements (parallel and antiparallel crossings of two possible signs), but also into higher composing elements that we call topological domains. All crossings belonging to a given topological domain are topologically equivalent to each other, and therefore a topological operation like an intersegmental passage performed at any of the crossings belonging to a given topological domain has the same topological consequence. Figure 3 shows that the right-handed 99 knot is composed of three topological domains. There is a domain with four right-handed parallel crossings, and a first intersegmental passage occurring at any of these crossings leads to the formation of the 75 knot. Then, there is a domain with three parallel right-handed crossings, and the first passage at any of these crossings leads to the formation of the 73 knot. Finally, there is a domain with two right-handed antiparallel crossings, and the first intersegmental passage at any of these crossings leads to the formation of the 71 knot. In the case of the 99 knot drawn in figure 3, the crossings belonging to a given topological domain form an uninterrupted row of double-arc regions on a minimal crossing diagram of this knot, giving a situation where topological domains are each composed of just one structural domain. This is not always the case. Figure 4 presents two different minimal diagrams of the 75 knot that can be converted into each other by the so-called flype move. Although flype moves (Adams 1994) can separate topological domains composed of uninterrupted rows of double-arc regions into two (or more) of such regions, this does not change the number of crossings belonging to each topological domain.
Frequently, knots with symmetrical diagrams have two or more identical topological domains. The topological difference between one or more domains of a given type can be well illustrated by the effect of two successive orientation-preserving smoothings performed on otherwise unchanged diagrams of a knot. Two such smoothings that are performed within the same antiparallel domain always lead to the creation of a disjoint diagram, while this is not the case when the two smoothings are redistributed into two identical but separate antiparallel domains (see figure 1a of electronic supplementary material).
Characters of topological domains (parallel or antiparallel) are defined with respect to the first passage occurring within a minimal diagram of a knot. During progressive simplification of knot diagrams, individual crossings can change their character from parallel to antiparallel (or contrary), but not the sign (figure 3, left). Domains containing such individual crossings are marked with asterisks in table 1 that lists the decompositions of alternating prime knots with up to 7 crossings into their parallel (P) and antiparallel (A) domains of different signs (in the electronic supplementary material, we have included a table listing topological domains of knots with up to 9 crossings). The numerical entries placed in the rows P or A in table 1 (and in a bigger table in the electronic supplementary material) indicate the number of crossings in composing parallel or antiparallel domains, respectively. The signs tell whether crossings in a given domain are positive or negative. Each asterisk, appending a given domain, indicates that one crossing within this domain changes the type but not the sign during the progressive simplification of the knot.
As previously mentioned, there are several possible minimal diagrams of a given knot. However, for all alternating knots analysed here, we did not observe any cases where different minimal diagrams of the same knot had different numbers of crossings in the corresponding topological domains. Unfortunately, the constancy of topological domains does not hold for minimal crossing diagrams of non-alternating knots. Each of these knots can have numerous different minimal crossing diagrams that differ in the number of crossings belonging to a given topological domain. However, all 8 crossing non-alternating knots 819, 820 and 821 have minimal crossing diagrams that can give rise to all of their predecessor knots upon strand passages occurring at the crossing points of these diagrams. These diagrams for knots 820 and 821 are different from the standard ones in the tables of knots (Rolfsen 1976; Adams 1994); therefore, we have presented them together with the indications of the knot types resulting from passages at each crossing (see figure 2 of electronic supplementary material). Alternating knots frequently have minimal crossing diagrams, where the crossings belonging to every topological domain composed of two or more crossings are grouped together into an uninterrupted row of double-arc fields, forming one structural domain (figure 3). In fact, if such a diagram exists for a given knot, it is usually presented in the standard tables of knots (Rolfsen 1976; Adams 1994). The entries in table 1 (and its extended version in the electronic supplementary material) are based on the analysis of minimal diagrams from such tables with the exception of non-alternating knots 820 and 821, for which we have used the diagrams shown in the electronic supplementary material. The type of the domain (parallel or antiparallel) is easy to define for domains composed of double-arc regions (figure 3). However, the assignment of the domain type for topological domains composed of crossings that do not belong to double-arc regions is more complex. In the electronic supplementary material, a more thorough explanation is provided how one can decide whether crossings not belonging to double-arc regions have parallel or antiparallel character.
Analysis of table 1 (and its extended version in the electronic supplementary material) reveals some simple rules of knots composition in the case of alternating knots, which are as follows.
All knots have at least one antiparallel crossing for each represented sign.
Parallel crossings exist as pairs, i.e. if a knot has parallel crossings, it has an even number of them, although some crossings may only reveal their real character (parallel or antiparallel) during the progressive simplification of the diagram.
For all knots with up to 9 crossings, their decomposition into topological domains was unique.
(e) Knots' business cards
Although table 1 (and its extended version in the electronic supplementary material) allows us to distinguish different knots based on the decomposition of their minimal diagram, such tables do not inform us about the consequence of strand passages in each of the topological domains. Therefore, such tables do not convey the information about the kinship of a given knot with all its predecessors. Figure 5 presents, in a somewhat abstract way, several simple knots where crossings constituting topological domains are ‘extracted’ and placed in an arbitrary manner not reflecting the actual arrangement and connectivity of the domains in the minimal diagrams. Interestingly, this representation seems to maintain the essential information about the represented knots. The topological domains presented in the upper (positive) parts of the respective panels are composed of positive crossings (the drawn direction arrows form positive crossings; figure 1). The topological domains drawn in the lower (negative) parts of the panels are the negatives of the positive crossings (in literal and mathematical sense). In parallel domains composed of more than one crossing, the direction arrows run in the same direction, while the opposite is the case for antiparallel domains. Crossings represented as ‘vertical’ rectangles correspond to these crossings that contribute values ±10/7 to the writhe of the knot, depending on the sign of the crossing. Horizontal rectangles correspond to these crossings that contribute ±4/7 to the writhe of the knot. Domains composed of vertical and horizontal rectangles indicate that the character of one of the crossings reveals itself only during further steps of progressive simplification of the knot. The number and character of all crossings in each topological domain of a given knot is thus characterized. The knot types denoted to the right of every topological domain indicate the predecessor knots that are created by the strand-passage reaction (sign inversion) at any of the crossings in a given topological domain. The relative ratio between the numbers of crossings in different topological domains determines the kinship order of a given successor knot to its predecessors. In contrast to figure 2, that, for the reasons of simplicity, traced mainly the kinship between the knots that differ in their number of crossings by 2, the panels representing individual knots list all simpler (predecessor) knots that can be created out of the represented knot by an individual segment–segment passage. We have checked that all the predecessor knots were consistent with the tables listing which knots can be converted to each other by one intersegmental passage (Darcy & Sumners 1997; Darcy & Sumners 2000). The link type denoted on the left-hand side of a given topological domain indicates what type of links is created by orientation-preserving smoothing of any of the crossings belonging to a given topological domain (see figure 1 of the electronic supplementary material for examples of orientation-preserving smoothing operations). In most of the cases, it is not necessary to resort to checking the consequences of the orientation-preserving smoothing to determine which crossings belong to the same topological domain. However, in the case of the 41 knot, a strand passage at any of the four crossings leads to the formation of unknot, and this may be interpreted as if all the crossings belong to the same topological domain. In contrast to this, the orientation-preserving smoothing performed at any of the positive antiparallel crossings leads to the creation of left-handed Hopf link, while right-handed Hopf link is created by smoothing performed at any of the negative antiparallel crossings. Therefore, the orientation-preserving smoothing can be used to unambiguously determine which crossings belong to the same topological domain. The numbers in the upper right-hand corner of every panel give the predicted writhe values of the represented knots (i.e. the writhe of ideal geometric representation and the mean writhe of statistical ensemble of random knots of a given type). To operate with integer numbers, the writhe is given in convenient units of 4/7. In the upper left-hand corners, we propose a notation of individual knots. This notation is similar to that applied in table 1, and it conveys the information about the decomposition of individual knots into composing crossings and topological domains. The capital ‘A’ and ‘P’ letters denote topological domains with antiparallel and parallel apparent character of composing crossings and with all crossings of positive sign. The lower-case letters ‘a’ and ‘p’ denote the corresponding topological domains with negative crossings. The asterisks stand for the crossings that change their character during the progressive simplification of the knot. Brackets group together the structural domains that constitute a given topological domain, but cannot form a continuous double-arc region on a minimal crossing diagram of the knot. In the lower part, we have listed the number of topological domains (Nd) and different topological outcomes (Nt) resulting from a smoothing of crossings in the minimal diagram. We have also listed the average number of crossings the knot loses after a random strand-passage (down right) or smoothing reaction (down left) occurring with equal probability at each crossing of a standard diagram of a given knot. We call this number the average relaxation value (Ar). Nd, Nt and Ar seem to be invariants for alternating knots and can be easily determined on any minimal crossing diagram of a given alternating knot.
Although a minimal diagram of a knot is its unique determinant, it could be compared to a picture of a person that at sufficient resolution is also its unique determinant. However, in our professional contacts, we prefer to operate with business cards as they convey more relevant information about a given person than its picture. The panels in figure 5 constitute business cards of knots and convey the information about all crossings building a given knot, the number of topological domains in a knot, the relatedness to the predecessors of this knot, etc. In the electronic supplementary material (figures 3–10), we have placed business cards of more knots, including all prime alternating knots with up to 9 crossings. For a better readability, we replaced the parallel crossings by square symbols and the antiparallel ones by circles and noted only the types of knots created by the strand passages within each topological domain.
We have presented the new way of classifying all the knot types that takes into account the natural relationship between various knot types resulting from random intersegmental passages within knots with up to 9 crossings.
This work was supported by Swiss National Science Foundation grants 3152-068151 and 3100A0-103962. We thank Profs Isabel Darcy, Klaus Ernst, Eric Rawdon and Mariel Vazquez for their comments on the manuscript.