A Topological Invariant to Predict the Three-Dimensional Writhe of Ideal Configurations of Knots and Links

A topological invariant to predict the three-dimensional writhe of ideal configurations of knots and links

Corinne Cerf* and Andrzej Stasiak†‡

*De´partement de Mathe´matique, Universite´Libre de Bruxelles, B-1050 Bruxelles, Belgium; and †Laboratoire d’Analyse Ultrastructurale, Baˆtiment de Biologie, Universite´de Lausanne, CH-1015 Lausanne-Dorigny, Switzerland

Communicated by Nicholas R. Cozzarelli, University of California, Berkeley, CA, February 3, 2000 (received for review November 16, 1999) We present herein a topological invariant of oriented alternat- 2D writhe and the 3D writhe are geometrical notions, because ing knots and links that predicts the three-dimensional (3D) they depend on the configuration. writhe of the ideal geometrical configuration of the considered Let us now consider the minimum of the 2D writhe over all knot͞link. The fact that we can correlate a geometrical property possible projections of all possible configurations of a link. The of a given configuration with a topological invariant supports obtained number depends on the link type only and is thus a the notion that the ideal configuration contains important topological invariant of the link. It is the topological writhe, information about knots and links. The importance of the denoted by w, which is also called Tait number, because the concept of ideal configuration was already suggested by the notion first appeared in the work of P. G. Tait (9). The good correlation between the 3D writhe of ideal knot configu- topological writhe is very simple to compute for alternating links rations and the ensemble average of the 3D writhe of random (links where the overcrossings and undercrossings alternate configurations of the considered knots. The values of the new along each component): it is the 2D writhe of any projection of invariant are quantized: multiples of 4͞7 for links with an odd the link without nugatory crossing (a nugatory crossing is a number of components (including knots) and 2͞7 plus multiples crossing that can be removed from the projection just by rotating of 4͞7 for links with an even number of components. a part of the link). Let us call such a projection a standard projection. The notions of writhe are crucial in knot theory and Ideal Configurations of Knots and Links in its applications to molecules like DNA, which can be knotted and catenated, because a nonzero writhe detects chirality (geo- knot type (or simply ‘‘knot’’) is a class of non-self- metrical chirality or topological chirality depending on whether A intersecting closed curves in three-dimensional (3D) space w or Wr is considered; ref. 10). that are ‘‘equivalent,’’ i.e., that can be continuously deformed into each other without passing through themselves. Many Prediction of the 3D Writhe from a Standard Projection efforts have been made to find for each knot type a ‘‘canonical’’ From its definition, it is easy to see that the topological writhe configuration by minimizing a configuration-dependent energy of a link is an integer. We can describe it as a quantity that is function (e.g., see various energy functions in ref. 1). One of the quantized by steps of 1. On the contrary, the 3D writhe is a real simplest and best-behaved energy functions is the thickness number, because it results from an averaging. It has been energy of the knot (2, 3). By representing the knot by a closed observed, however, that the 3D writhe of ideal configurations of impenetrable cylindrical tube of uniform diameter whose axis is knots obtained by computer simulations is quantized too but not the knot, the associated thickness energy is the ratio of the length by steps of 1 (3, 11). For example (3), in the family of torus knots

of the tube axis (L) over the diameter of the tube (D). The knot (31,51,71, . . . ), each time two crossings are added, w changes by APPLIED

configuration at which the L͞D ratio reaches its global minimum steps of 2, whereas Wr of the corresponding ideal configurations MATHEMATICS has been termed ‘‘ideal’’ (3). Generalization to a link type, i.e., changes by steps of about 2.86; in the families of twist knots with (a class of) several closed nonintersecting curves called compo- even (41,61,81, . . . ) or odd (52,72,92, . . . ) crossing number, each nents, can be done by imposing that the diameters of all time two crossings are added, w changes by steps of 2, whereas component tubes be identical (4). Up to now, ideal configura- Wr of the corresponding ideal configurations changes by steps of tions for knots and links have been obtained by computer about 1.14 (see Fig. 1). It was suggested in ref. 12 that the steps simulations. in Wr could actually be rational numbers (i.e., 20͞7 and 8͞7, The ideal configuration is believed to constitute an essential respectively), such that the addition of each crossing contributes characteristic of knots and links. For example, the 3D writhe of 10͞7 in the first case and 4͞7 in the second case. By inspecting ideal knot configurations (which will be defined in the next standard projections of many links, we could extend this idea and paragraph) correlates very well with the ensemble average of the arrive at the following formula for alternating links: 3D writhe of random configurations of the considered knots PWr ϭ 10͞7w ϩ 4͞7w ϭ w ϩ 3͞7͑w Ϫ w ͒, [1] (3, 5, 6). x y x y 2D Writhe, 3D Writhe, and Topological Writhe where PWr stands for predicted 3D writhe of the ideal configuration of the link and w, wx, and wy are the topological writhe, Different notions of writhe are associated with oriented links the nullification writhe, and the remaining writhe of the link, (from now on, knots will be considered as special cases of links respectively. The nullification writhe and the remaining writhe with one component, and orientation will always be assumed). are defined as follows (13): transform a standard projection by Projecting a particular configuration of a link in a particular nullifying (or smoothing) successive crossings until the unknot is ϩ direction, each right-handed crossing is scored as 1 and each reached, while, at each step, preventing the diagram from Ϫ left-handed crossing is scored as 1 (e.g., see Fig. 1; ref. 7). The becoming disconnected. Notice that, on each elimination of a total of these is called the 2D writhe of the given projection of the given link configuration. The average of the 2D writhe over all possible directions of projection constitutes the 3D writhe of Abbreviations: 2D and 3D, two- and three-dimensional. the given link configuration and is denoted by Wr (8). Both the ‡To whom reprint requests should be addressed. E-mail: [email protected].

PNAS ͉ April 11, 2000 ͉ vol. 97 ͉ no. 8 ͉ 3795–3798 Downloaded by guest on October 2, 2021 Fig. 3. Correlation between the computed 3D writhe of the ideal configuration of alternating knots and links (Wr) and the predicted value (PWr). Alternating knots up to eight crossings are indicated by an open diamond (Wr data from ref. 5), and alternating two-component links up to four crossings are indicated by a filled diamond (Wr data provided by Ben Laurie, personal communication). It is to be observed that the PWr of knots Fig. 1. Families of knots, with crossing sign assignment. (Top) Family of are multiples of 4͞7 (vertical dotted lines) and that the PWr of two- torus knots. (Middle) Family of twist knots with even crossing number. component links are equal to 2͞7 plus multiples of 4͞7 (between two (Bottom) Family of twist knots with odd crossing number. Wr is the 3D vertical dotted lines). writhe of the ideal configuration obtained by simulation. (Note that for each chiral knot, we chose the enantiomorph having a positive topological writhe. For all the knots represented in this figure, this enantiomorph corresponds to the mirror image of the enantiomorph usually presented in property of a given link configuration starting from topological reference tables; ref. 7.) invariants of the link supports the notion that the ideal configuration contains essential information about alternating links. The reader might be interested to compare this way of predicting crossing, the open ends are spliced in such a way that the Wr with the one presented in ref. 12. Dylan Thurston (personal directionality of the knot contour is maintained (14). Then, wx communication) pointed out to us that PWr can also be ex- is the sum of the signs of the nullified crossings, and wy is the sum pressed as w ϩ 3͞7(no. white Ϫ no. black), where no. white and ϩ ϭ of the signs of the remaining crossings, with wx wy w.An no. black are the numbers of white and black regions in a example is shown in Fig. 2. checkerboard coloring of the link. PWr is an invariant of oriented alternating links, because w, wx, About the 3͞7 Factor and wy are invariants of oriented alternating links (13). More- over, its correlation with the actual Wr of the ideal configuration A natural question to ask about the formula of PWr is whether of alternating links obtained by simulations is remarkably good the 3͞7 factor is just an ad hoc adjustment that could as well be (see Fig. 3). The fact that we can accurately predict a geometrical replaced with 0.42 or 0.43 or whether it has some deeper

Fig. 2. Example of calculation of the nullification writhe (wx) and the remaining writhe (wy) of a knot. Successive crossings are nullified until the unknot is reached, while, at each step, the diagram is prevented from becoming disconnected. The open ends arising after elimination of crossings are spliced in such a way that directionality along the knot contour is maintained. Then, wx is the sum of the signs of the nullified crossings, and wy is the sum of the signs of the remaining crossings, with wx ϩ wy ϭ w, the topological writhe.

3796 ͉ www.pnas.org Cerf and Stasiak Downloaded by guest on October 2, 2021 However, some equations do lead to the solution x ϭ 3͞7. Actually, all equations that we tried that do not lead to an indeterminacy lead to that solution. Two examples are given hereafter, when identifying the PWr of knots 72 and 82 and the PWr of knots 81 and 98 (see Fig. 4): 7 ϩ x͑2 Ϫ 5͒ ϭ 4 ϩ x͑4 Ϫ 0͒ 3 x ϭ 3͞7, [3] 4 ϩ x͑0 Ϫ 4͒ ϭ 1 ϩ x͓2 Ϫ ͑Ϫ1͔͒ 3 x ϭ 3͞7. [4] In conclusion, if one admits a perfect quantization of the 3D writhe of ideal configurations of alternating links, the factor in the formula of PWr has to be exactly 3͞7. Quantization of the 3D Writhe Observing and trying to understand the quantization of the 3D writhe of ideal knot configurations has been a hot subject during recent years (3, 5, 6, 11, 12). Let us analyze the ϭ ͞ ϩ ͞ consequences of the formula PWr 10 7 wx 4 7 wy on the quantization of the Wr of ideal configurations of alternating links. The largest common divider of 10͞7 and 4͞7is2͞7; thus, PWr must be quantized by steps of 2͞7. Actually, it is possible to make a stronger statement, by invoking the fact that wx is even for an alternating link with an odd number of components and odd for an alternating link with an even number of components (13). It follows that PWr of alternating links with an odd number of components (including knots) is a multiple of 4͞7, whereas PWr of alternating links with an even number of components is equal to 2͞7 plus a multiple of 4͞7. All observed values of Wr are in accordance with these predictions (see Fig. 3). Fig. 4. Some pairs of knots have observed values of Wr close to each other. Wr is the 3D writhe of the ideal conﬁguration obtained by simulation. (Note that, for Discussion each chiral knot, we chose the enantiomorph having a positive topological The topological invariant of alternating knots and links described writhe. For some knots, this enantiomorph corresponds to the mirror image of in this paper, PWr, is altogether very easy to calculate from a the enantiomorph usually presented in reference tables; ref. 7). standard projection, and it provides a testing ground for the powerful conjecture that it correlates with geometrical and physical properties of knots and links. This conjecture predicts justification. A partial answer to this question is the following. with great accuracy the 3D writhe of an ideal knot configuration Some observed values of 3D writhe for the ideal configuration or, equivalently, the 3D writhe averaged over an ensemble of of different links are very close to each other (5, 11), and we are random knot configurations. tempted to make the hypothesis that the discrepancies arise from Until now, most simulations of 3D writhe have been done on the intrinsic limitations of the simulations. For example, it is an

knots, and some have been done on two-component links. We APPLIED open problem whether the configuration obtained by minimizing

suggest herein to extend the simulations to three- and four- MATHEMATICS the thickness energy of a link corresponds to a global minimum. component alternating links, which would allow our predictions Another example of limitation is the necessity to represent the to be checked experimentally relative to the 3D writhe of their curves by a succession of straight segments. Making the hypoth- ideal configuration and the confirmation of their fascinating esis that a ‘‘perfect’’ simulation would yield exactly the same 3D property of quantization. writhe for the links whose observed 3D writhes are close to each As we already mentioned, PWr is invariant for alternating other, we would like that the corresponding predicted 3D writhes knots and links only. More work is still needed to try to find an PWr be equal. Imposing this condition gives rise to a series of invariant of nonalternating as well as alternating knots and links ͞ equations in which the factor 3 7 is replaced with a variable x. that would correlate with the 3D writhe of ideal knots and links Most of these equations lead to an indeterminacy, the left- and configurations. right-hand sides being identical. As an example, imposing that Let us finally express the wish that this work might help and the PWr of knots 31 and 76 are equal (see Fig. 4) gives rise to the motivate the effort to find an analytical solution to the ideal equation: configuration of knots and links. ϩ ͑ Ϫ ͒ ϭ ϩ ͑ Ϫ ͒ 3 x 2 1 3 x 2 1 . [2] We thank Piotr Pieranski for numerous discussions on the quantization of writhe, Dylan Thurston for sharing his ideas about PWr, Ben Laurie [Note that for each chiral link, we choose the enantiomorph for providing us with his unpublished Wr data on two-component links, that has a positive w. The equation would be the same (each and Jacques Dubochet and Claude Weber for their interest in the project. term would be multiplied by Ϫ1) had we chosen the other This work was supported in part by Swiss National Foundation Grant enantiomorph.] 31-42158.94 to J. Dubochet and A.S.

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