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Computing the Writhing Number of a Polygonal Knot

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Computing the Writhing Number of a Polygonal Knot Computing the Writhing Number of a Polygonal Knot ¡ ¡£¢ ¡ Pankaj K. Agarwal Herbert Edelsbrunner Yusu Wang Abstract Here the linking number, , is half the signed number of crossings between the two boundary curves of the ribbon, The writhing number measures the global geometry of a and the twisting number, , is half the average signed num- closed space curve or knot. We show that this measure is ber of local crossing between the two curves. The non-local related to the average winding number of its Gauss map. Us- crossings between the two curves correspond to crossings ing this relationship, we give an algorithm for computing the of the ribbon axis, which are counted by the writhing num- ¤ writhing number for a polygonal knot with edges in time ber, . A small subset of the mathematical literature on ¥§¦ ¨ roughly proportional to ¤ . We also implement a different, the subject can be found in [3, 20]. Besides the mathemat- simple algorithm and provide experimental evidence for its ical interest, the White Formula and the writhing number practical efficiency. have received attention both in physics and in biochemistry [17, 23, 26, 30]. For example, they are relevant in under- standing various geometric conformations we find for circu- 1 Introduction lar DNA in solution, as illustrated in Figure 1 taken from [7]. By representing DNA as a ribbon, the writhing number of its The writhing number is an attempt to capture the physical phenomenon that a cord tends to form loops and coils when it is twisted. We model the cord by a knot, which we define to be an oriented closed curve in three-dimensional space. We consider its two-dimensional family of parallel projections. In each projection, we count © or for each crossing, depending on whether the overpass requires a counterclock- wise or a clockwise rotation (an an angle between 0 and ) to align with the underpass. The writhing number is then the signed number of crossings averaged over all parallel projec- tions. It is a conformal invariant of the knot and useful as a Figure 1: Circular DNA takes on different supercoiling conforma- measure of its global geometry. tions in solution. The writhing number attracted much attention after the relationship between the linking number of a closed ribbon and the writhing number of its axis, expressed by the White axis measures the amount of supercoiling, which character- formula, was discovered independently by Calug˘ areanu˘ [9], izes some of the DNA’s chemical and biological properties Fuller [21], Pohl [27], and White [33]: [5]. (1) © This paper studies algorithms for computing the writhing number of a polygonal knot. Section 2 introduces back- Research by all three authors is partially supported by NSF under grants ground work and states our results. Section 3 relates the CCR-00-86013 and EIA-9972879. Work by the first author has also been supported by by NSF grants EIA-98-70724, EIA-01-31905, and CCR-97- writhing number of a knot with the winding number of its 32787, and by a grant from the U.S.–Israel Binational Science Foundation. Gauss map. Section 4 shows how to compute the writhing Research by the second author is also partially supported by NSF under number in time less than quadratic in the number of edges of grant CCR-97-12088. the knot. Section 5 discusses a simpler sweep-line algorithm Department of Computer Science, Duke University, Durham, North Carolina, USA. email: pankaj, edels, [email protected] and presents initial experimental results. Section 6 concludes Raindrop Geomagic, Research Triangle Park, North Carolina, USA. the paper. 2 Prior and New Work If the smooth knot is approximated by a polygonal knot, we can turn the right hand side of (3) into a double sum and In this section, we formally define the writhing number of a approximate the writhing number of the smooth knot [6, 26]. knot and review prior algorithms used to compute or approx- This can also be done in a way so that the double sum gives imate that number. We conclude by presenting our results. the exact writhing number of the polygonal knot [4, 24, 32]. ¦¥¨§ © ¥ Alternatively, we may base the computation of the Definitions. A knot is a continuous injection ¢¡¤£ § © writhing number on the directional version of the White for- % % © ?£ or, equivalently, an oriented closed curve embedded in . mula, for . Recall that both the We use the two-dimensional sphere of directions, £ , to rep- § © linking number and the twisting number are defined over the resent the family of parallel projections in . Given a knot two boundary curves of a closed ribbon. Similar to the defi- % £ and a direction , the projection of is an oriented, nition of , the directional twisting number, , is possibly self-intersecting, closed curve in a plane normal to defined as half the sum of crossings between the two curves, . We assume to be generic, that is, each crossing of in each counted as © or as described in Figure 2. We get the direction is simple and identifies two oriented intervals (1) by integrating over £" and noting that the linking number along , of which the one closer to the viewer is the over- does not depend on the direction. This implies pass and the other is the underpass. We count the crossing as © if we can align the two orientations by rotating the over- @BA A pass in counterclockwise order by an angle between and © ¦! (4) . Similarly, we count the crossing as if the necessary rotation is in clockwise order. Both cases are illustrated in Figure 2. The Tait or directional writhing number of in To compute the directional and the (average directional) twisting numbers, we expand to a ribbon, which amounts to constructing a second knot that runs alongside but is dis- joint from . Expressions for these numbers that depend on how we construct this second knot can be found in [24]. Le Bret [25] suggests to fix a direction and define the sec- +1 −1 ond knot such that in the projection it runs always to the left @ of . In this case we have and the writhing Figure 2: The two types of crossings when two oriented intervals number is the directional writhing number for minus the intersect. twisting number. A third approach to computing the writhing number is the direction , denoted as , is the sum of crossings based on a result by Cimasoni [16], which states that the counted as © or as explained. The writhing number writhing number is the directional writhing number for a is the averaged directional writhing number, taken over all fixed direction , plus the average deviation of the other di- directions £ , rectional writhing numbers from . By observing that BA A C is the same for all directions in a cell of the "!# (2) decomposition of £ formed by the Gauss maps and (also referred to as the tangent indicatrix or tantrix in the We note that a crossing in the projection along also ex- literature [14, 29]), we get $ ists in the opposite direction, along , and that it has the % % E E $ same sign. Hence , which implies that © 8 JILK ( (5) -DFEHG the writhing number can be obtained by averaging the direc- tional writhing number over all points of the projective plane MA A E $ *) or, equivalently, over all antipodal points pairs &' #( of the where is for any one point in the interior of E K C sphere. C , and is the area of . If applied to a polygonal knot, all three algorithms take Computing the writhing number. Several approaches to time that is at least proportional to the square of the number computing the writhing number of a smooth knot exactly or of edges in the worst case. approximately have been developed. Consider an arc-length ¥ ¥¨§ © + + parameterization ¨¡¤£ , and use and to denote Our results. We present two new results. The first result ¥ the position and the unit tangent vectors for ,-.£ . The can be viewed as a variation of (4) and a stronger version of A following double integral formula for the writhing number (5). For a direction N£" not on and not on , let %MA can be found in [27, 31]: be its winding number with respect to and . As %MA 0/ 0/21 65 5:9 explained in Section 3, this means that and wind A +43 (7 +£ 8 © !=, !#> times around . ; ; (3) 5 < + 2 THEOREM A. For a knot and a direction , we have Critical directions. We specify a polygonal knot by ¥ ¥ ¥§¦ (¨¥ ( (©¥ the cyclic sequence of its vertices, in § © %BA A % ¥ ¡ ¤ , ¥§ . We use indices modulo and write ¦! £© ; ; ¥ ¥ ¡ ¡ ¥¥§ ¥ ¥ ¥ for the unit vector along the edge . © § £ Note that , is also a direction in and a point in . Any ¥ ¡ , Observe the similarity of this formula with (4), which sug- two consecutive points , and determine a unique arc, gests that the winding number can be interpreted as the direc- which, by definition, is the shorter piece of the great cir- ¥ ¥ ¦¤(F, ( ( , tional twisting number for a ribbon one of whose two bound- cle that connects them. The cyclic sequence , £ ary curves is . We will prove Theorem A in Section 3. We thus defines an oriented closed curve in . We also need will also extend the relation in Theorem A to open knots and the antipodal curve, , which is the central reflection of give an algorithm that computes the average winding number through the origin. in time proportional to the number of edges. Our second re- sult is an algorithm that computes the directional writhing number for a polygonal knot in time sub-quadratic in the number of edges. ¤ THEOREM B. Given a polygonal knot with edges and £ a direction , can be computed in time ¥§¦ ¨ ¡£¢ ¤ O ¤ , where is an arbitrarily small positive con- stant.
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