## Abstract

The knot nomenclature in common use, summarized in Rolfsen's knot table [Rolfsen (1990) Knots and Links, American Mathematical Society], was not originally designed to distinguish between mirror images. This ambiguity is particularly inconvenient when studying knotted biopolymers such as DNA and proteins, since their chirality is often significant. In the present article, we propose a biologically meaningful knot table where a representative of a chiral pair is chosen on the basis of its mean writhe. There is numerical evidence that the sign of the mean writhe is invariant for each knot in a chiral pair. We review numerical evidence where, for each knot type *K*, the mean writhe is taken over a large ensemble of randomly chosen realizations of *K*. It has also been proposed that a chiral pair can be distinguished by assessing the writhe of a minimal or ideal conformation of the knot. In all cases examined to date, the two methods produce the same results.

- average crossing number
- chirality
- DNA knot
- knot table
- lattice knot
- writhe

## Knot tables and knot chirality

The chirality of a knot *K*, i.e. the difference between *K* and its mirror image *K**, plays an important role in the study of biopolymers. Biopolymers such as DNA and proteins are often knotted and linked. Experimentalists use DNA knots as probes for characterizing biophysical properties of DNA in solution and in confinement (e.g. [1]), and for analysing enzymatic actions (reviewed in [2]). Biological processes yielding knotted and linked (catenated) DNA often bias one chirality over the other. Examples include type II topoisomerases, which are believed to interact with DNA in a chiral manner (e.g. [3–6]), and DNA packing in bacteriophages [7]. When studying knots and links, it is important for the scientific community to use consistent nomenclature. However, the current knot tables are ambiguous when it comes to distinguishing a knot from its mirror image. The knot table proposed in Figure 1 offers a consistent way to identify chiral knots on the basis of their writhe, which is defined in the background section below.

Scientists needing to visually verify the identity of a given knot typically refer to the knot table in Rolfsen's *Knots and Links* [8]. The Rolfsen table [8] labels knots using both Conway's notation [8a] and an extension of the notation previously used by Alexander and Briggs [9]. Conway's notation [8a] built on early work by Tait [9a]. Since then, many other books, websites and software have borrowed from this table either directly or indirectly ([10,11], and http://www.indiana.edu/~knotinfo and http://katlas.org/). Often, the choice for the representative of a chiral pair is not consistent across the different tables.

The Alexander–Briggs notation [9] is compact and easy to use as it specifies a hierarchy guided by the minimal crossing number (defined in the background section below). However, it suffers from a few disadvantages, for example the order within fixed crossing number is more or less arbitrary. A disadvantage that is the focus of the present paper is that this notation makes it impossible to distinguish a knot from its mirror image.

## Writhe-based knot table

In [12], the authors proposed a new naming convention that modifies the Alexander–Briggs notation [9] and systematically distinguishes knots within each chiral pair (for ten and fewer crossings). This new convention uses two ideas related to the writhe of a polygonal chain. The writhe is a geometrical property, which provides a measure of a chain's entanglement complexity and chirality. More detailed definitions of the writhe are given in the next section. The ideas used in the writhe-based convention were based on earlier studies, notably [13] for chains in ℝ^{3} and [14] for lattice chains.

The first idea is that, whereas knots of a given type can be realized by infinitely many conformations, the statistical properties of large unbiased ensembles of conformations of a given knot type tend to be consistent. This idea has been explored broadly in the literature. For example, properties such as the average crossing number, the radius of gyration or the writhe have been studied for large ensembles of knots of the same type, generated using algorithms which sample the space of conformations uniformly (e.g. [12–21]). Notably, the interplay between knotting and writhe in polygons has been studied (e.g. [22,23]). It was proposed in [12] that, because the sign of the overall mean writhe (defined below) depends on the chirality of the chosen knot type, the sign could be used to modify Rolfsen's table [8] in order to distinguish mirror images objectively. The second idea involves certain minimal conformations that seem to have simpler geometry than the general conformation of a given knot type. Examples include: knots with minimal length to diameter ratio, called maximally inflated knots [24], ideal knots [13], and knots with minimal rope length [26]. In the simple cubic lattice, we consider minimal edge polygons of a given knot type to be ‘ideal’ shapes for that knot type. Minimal lattice knots are discussed in [27–29]. It has been reported that the mean writhe of maximally inflated knots and minimal lattice knots tends to be similar to the mean writhe computed over large ensembles of conformations [12–14,23,25,29,30].

Figure 1 shows all of the prime knots with eight or fewer crossings with the correct chirality according to the writhe-guided naming convention from [12]. A more complete table of knots with up to ten crossings can be obtained from M.V. upon request.

## Mathematical background: knots, chirality and writhe

A knot is a closed curve in ℝ^{3} that does not intersect itself (reviewed in [10]). Two knots are considered equivalent, or of the same knot type, if there is a way of continuously deforming one knot on to the other without causing it to cross itself. All physical knots can be realized as equilateral polygons. Define the set of realizations of a given knot type *K* as the equivalence class consisting of all equilateral polygons in ℝ^{3} having knot type *K*.

A knot invariant is a topological property that remains constant within the set of realizations of each knot type. Most invariants are not in one-to-one correspondence with the set of knot types, but will often describe interesting topological or geometric features common to all realizations of the same type. Two examples already mentioned are minimal crossing number and chirality. A knot is achiral if it is of the same knot type as its mirror image, and chiral otherwise.

Crossing number, ACN (average crossing number) and writhe are three closely related concepts which give slightly different information about the topological and geometrical complexity of a knot. All three are related to the idea of regular diagram for a knot. A regular diagram is a planar representation of a knot where every crossing is a double crossing and which distinguishes between under- and over-crossings. Knots are studied through their regular diagrams. We define the crossing number of the diagram as the total number of crossings, and the ACN of a knot as the average of the crossing number taken over all possible diagrams. The minimal crossing number of a knot type *K*, which is a topological invariant, is the smallest crossing number for any diagram for any realization of *K*. The minimal crossing number is the first part of the Alexander–Briggs notation [9]. Knots are denoted by their minimal crossing number *n* with a subscript depending on the order of the table (Figure 1). We refer to knots with minimal crossing number *n* as *n*-crossing knots. There is only one way that a knot can have zero minimal crossing number, and that is if the knot is simply a ring. This knot is called the trivial knot, or the unknot, and is denoted by 0_{1}. There are no two-crossing knots, but there are two three-crossing knots, one of which is the mirror image of the other. Both of these knots are referred to as trefoils and denoted as 3_{1} and 3_{1}*. There is exactly one four-crossing knot, because that knot is achiral. This is often called the figure-of-eight knot and is denoted by 4_{1}. Counting mirror images, there are four five-crossing knots denoted by 5_{1}, 5_{1}*, 5_{2} and 5_{2}*. Beyond this, the number of knots with a given crossing number is known to increase exponentially. There are five six-crossing knots, 14 seven-crossing knots, 37 eight-crossing knots, 98 nine-crossing knots and 317 ten-crossing knots.

One can assign an arbitrary orientation to any given knot *K* (Figure 2). Following this orientation, each crossing in a diagram of *K* can be assigned a (+1) or a (−1) based on the convention illustrated in Figure 2. The projected writhe of the knot diagram is the signed sum of the crossings. The writhe of the knot *K* is the average of the projected writhes taken over all possible regular diagrams of *K*. The mean writhe of *K* of length *n* is the writhe averaged over all *n*-edge realizations of the knot *K*. The ACN and writhe are geometric quantities; they can be changed by adding twists to a knot. The minimal crossing number, on the other hand, depends only on knot type. Crossing number and projected writhe are illustrated in Figure 2.

In [13], simulations of knotted DNA chains 1800 and 5400 bp long were conducted and the mean writhe of each of the ensembles was calculated. The authors observed, “the writhe values for different types of knots seem to be independent of DNA size”. In [12], the authors conjectured that the sign of the mean writhe of the ensemble of all polygons of fixed length and knot type is a topological invariant of chiral knots. Numerical data were provided to support this conjecture. A knot nomenclature was proposed where, given a chiral pair (*K*_{1}, *K*_{2}) of a fixed knot type *K*, the knot *K _{i}* is called

*K*if the mean writhe of

*K*is positive, otherwise it is called the mirror image of

_{i}*K*and denoted by

*K**.

The ACN is a geometric rather than topological property, but the average of ACN taken over a random distribution of knots of fixed type has been considered a knot invariant which is linearly related to the sedimentation coefficient of different types of DNA knots of equal length [31]. Similarly, Portillo et al. [12] proposed to treat the sign of the mean writhe as an invariant of chiral knots. Taking a mirror image does not change the crossing number of the diagram, but it switches the signs of each crossing and thus changes the sign of the writhe. Thus ACN will be the same for a conformation and its mirror image, but writhe will be of opposite sign. Thus the expected writhe of an unbiased distribution of realizations of an achiral knot must be strictly zero. Only chiral knots can have non-zero expected writhe, and the value for the two knots in a chiral pair will always have opposite sign.

## Computational methods

The simplest knot invariant is the set of all realizations of a knot type. This is an observation that seems almost too trivial to mention, but consider that subsets of the set of all realizations will also be invariants, and thus so will probability distributions on those sets. These subsets can be concrete and can be studied using stochastic methods. For example, the set of simple cubic lattice polygons of given length is a finite set.

Consider two polygons the same if one can be moved into the same position as the other by translation or rotation. Then, in the simple cubic lattice, there is exactly one polygon of length four. That polygon is the unit square, which is a realization of the unknot 0_{1}. Thus the set of all length-four simple-cubic-lattice realizations of the unknot has one element, modulo translation and rotation. Similarly, for each given length *n*, there is a finite number of polygons of length *n* in the simple cubic lattice, and thus finitely many realizations of any given knot type *K* of length *n*. The number of realizations grows exponentially as *n* goes to infinity, but remains finite for any fixed length.

There are many algorithms designed to produce statistical ensembles of three-dimensional polygons. Two notable examples are the ergodic crankshaft algorithm acting on polygonal chains in ℝ^{3} [32,33], and the BFACF algorithm acting on polygons in the simple cubic lattice ℤ^{3} [34,35]. The BFACF algorithm generates Markov chains of lattice polygons, and has the useful property that its ergodicity classes are exactly the sets of realizations of knot types [36]. In other words, every conformation generated by BFACF from an initial conformation of knot type *K* will have exactly the same knot type. Thus the BFACF algorithm is a good tool for generating large ensembles of lattice chains of fixed knot type. Properties of lattice polygons, and the BFACF algorithm are reviewed in detail in [37].

Using standard measures of independence, it is possible to generate large ensembles of essentially independent knots of fixed type. Statistical properties of these ensembles can be quantified. In the present article, we focus on the writhe. Portillo et al. [12] showed that, for all knots with eight crossings or fewer, the sign of the mean writhe of chiral knots was consistent for lengths ranging from 75 to 350. Furthermore, these signs unambiguously distinguished between the chiral knots with eight or fewer crossings and their corresponding mirror images. A smaller set of experiments was run for equilateral random polygons in three-dimensional space, randomized using the crankshaft algorithm. Results off-lattice also supported the conjectured invariance of the sign of the mean writhe. We extend the lattice studies to include all knots of ten or fewer crossings, with approximately the same precision. Results are reported in Table 1.

In [12,14], it was found that for each knot type *K*, the mean writhe of minimal lattice knots was very similar to the mean writhe taken over BFACF-generated ensembles of knots for each of the lengths studied. In particular, the sign of the mean writhe remains consistent throughout. Thus the convention of Portillo et al. [12] is consistent whether the sign of the writhe comes from the average of a large ensemble of lattice knots, or from the small set of minimal lattice polygons for each knot type. Comparisons for knots with up to ten crossings are reported in Table 1. Minimal lattice knots were obtained using BFACF as described in [27–29].

## Future work

In the present article, we extend the results from [12] to all prime knots with ten or fewer crossings. The results are still consistent with the conjecture on the invariance of the sign of the average writhe. We present a writhe-based knot table where the chirality of the knot is clearly specified. Detailed mathematical work needs to be done to determine whether the conjecture holds in general, and to investigate whether the mean writhe can be predicted from a minimal diagram for any knot or link, as proposed in [38]. Such mathematical validation would decrease the need for computer simulations.

It is also pertinent to add a comment about links with two or more components. A table of alternating links can be found in [39] and at http://at.yorku.ca/t/a/i/c/31.htm. The problem of dealing with several component links rather than knots is more delicate. Links sometimes have components which can be exchanged without changing the topology, and sometimes have components which in themselves are knotted. In addition, relative orientation of the components as parametrized curves can make a much bigger difference for links. A detailed study of the symmetry of links with eight or fewer crossings is provided in [40]. Whereas most knots have at most two symmetries consisting of image and mirror image, multiple component links can have many more symmetries, all requiring some form of disambiguation [41]. Some work has already been done in the direction of categorizing links with consistent orientations [39], but much remains to be done. We expect that invariants from stochastic models and minimal conformations will help to resolve ambiguities in these cases as well. A detailed study of links and a proposed link nomenclature will be included in a future publication.

## Funding

Support for this work came from the National Institutes of Health [grant number 2P20MD000544-06 (to R.S. and M.V.)], and from the National Science Foundation [grant numbers DMS0920887 (to M.V.) and DMS1057284 (to R.B.)].

## Acknowledgments

We thank the conference organizers for the opportunity to present our work. We thank Stu Whittington and Chris Soteros for very helpful discussions, and Juliet Portillo for all the background work on computing mean writhe.

## Footnotes

Topological Aspects of DNA Function and Protein Folding: An Independent Meeting held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., 3–7 September 2012, as part of the Isaac Newton Institute Programme Topological Dynamics in the Physical and Biological Sciences (16 July–21 December 2012). Organized and Edited by Andrew Bates (University of Liverpool, U.K.), Dorothy Buck (Imperial College London, U.K.), Sarah Harris (University of Leeds, U.K.), Andrzej Stasiak (University of Lausanne, Switzerland) and De Witt Sumners (Florida State University, U.S.A.).

**Abbreviations:**
ACN, average crossing number

- © The Authors Journal compilation © 2013 Biochemical Society