The Self-Linking of Torus Knots

Transcription

1 The Self-Linking of Torus Knots Edgar J. Fuller, Jr. Department of Mathematics, West Virginia University, Morgantown, WV Abstract. Torus knots are a widely studied class of space curves, convenient because of the surface on which they lie and the natural way in which they are defined. Costa [3], Little [6] and Romero-Fuster [8] have studied curvature and torsion properties of space curves previously. In the present work, some geometric properties of these knots are presented. The main result shows that the self-linking invariant for a (p, q) torus knot changes as the rigid geometric structure of the torus on which the curve lies is changed. It has been shown by Banchoff [1] that (p, q) curves on the flat torus in S 3 have constant self-linking invariant. Keywords: torus knots, self linking, curvature Mathematics Subject Classifications (2000): 57M25, Introduction In this section we recall some of the basic geometric properties of curves in R 3 as well as some less well known geometric invariants. See [9] or [10] for more comprehensive treatments. Throughout, a closed space curve will be taken to mean a closed, C, immersion of S 1 into R 3 unless otherwise noted. The domain parameter s will be used to represent a unit speed parameter and t an arbitrary one. Given a parametrization α : S 1 R 3 of such a closed space curve we define the curvature, κ, of α to be the norm of dt ds, namely κ(p) = d2 α (p). (1) ds2 Similarly, the torsion, τ of α is given by τ(p) = [α (p), α (p), α (p)] κ(p) 2 (2) where [,, ] denotes the scalar triple product. When dt ds is defined, i.e. dt ds when κ 0, define the Frenet frame at a point p = α(t 0 ) to be the triad of vectors T = dα ds N = dt ds dt ds B = T N (3) (4) c 2003 Kluwer Academic Publishers. Printed in the Netherlands. torusslv2.tex; 10/01/2003; 2:05; p.1

2 2 the unit tangent, unit principal normal and binormal vectors, respectively. A useful characterization of the curvature κ of a curve α lying on a surface S with second fundamental form II(, ) is in terms of the normal curvature, κ n, and geodesic curvature, κ g. These components can be computed as, where α(t) is not necessarily parametrized by arclength, κ 2 = κ 2 2 n + κ g (5) κ n = II(α (t), α (t)) ( ) dt 2 ( ) dt 3 κ g = [n(t), α (t), α (t)]. ds ds (6) The derivatives α and α are the first and second vector valued derivatives of α with respect to a parameter t, and n(t) is the normal to the surface S restricted to the curve α(t). The degeneracy and non-degeneracy of curvature and torsion will play an important role in what follows so it is convenient to have alternative criteria for their vanishing. Define the m-tangent space to α at p to be the linear span of the first m vector derivatives of α. It is denoted T m (α, p) = span{ d1 α dt 1,..., dm α dt m } (7) In this notation, κ(p) = 0 dim(t 2 (α, p)) = 1 and τ(p) = 0 dim(t 2 (α, p)) = dim(t 3 (α, p)) = 2 for any immersed space curve α. Note that these linear algebra constructions are independent of whether or not α is unit-speed parametrized. We say that α has an inflection of order k at p if T 1 (α, p) = T 2 (α, p) =... = T k+1 (α, p) T k+2 (α, p) (8) We can sometimes write T m (α) for T m (α, p) when the point in question is clear from the context or is arbitrary. Spatial inflections of order k 1 will be of interest as they correspond to points of zero curvature for space curves. Planar inflections and how they correspond to spatial inflections form the cornerstone of the analysis of space curves using planar knot diagrams. 2. Curvature Properties This section focuses on (p, q) torus curves, where gcd(p, q) = 1 in order to ensure they are closed but possibly unknotted space curves lying on a torus of revolution embedded in R 3 gotten by rotating a circle of torusslv2.tex; 10/01/2003; 2:05; p.2

3 radius b about the x 3 -axis with its center lying on the unit circle in the x 1 x 2 -plane. A parametrization for such a torus may be taken to be x(u, v) = ((1 + b cos(v)) cos(u), (1 + b cos(v)) sin(u), b sin(v)) (9) and on the surface T, we may take a (unitarized) moving frame corresponding to this parametrization to be x u = (sin(u), cos(u), 0) x v = (sin(v) cos(u), sin(v) sin(u), cos(v)) n = (cos(u) cos(v), sin(u) cos(v), sin(v)) These yield in turn a parametrization for (p, q) curves which will be used throughout given by α(t) = ((1 + b cos(qt)) cos(pt), (1 + b cos(qt)) sin(pt), b sin(qt)). (10) and the corresponding moving frame along α (inherited from the above moving frame by sending u to pt and v to qt ) given by: x u (t) = (sin(pt), cos(pt), 0) x v (t) = (sin(qt) cos(pt), sin(qt) sin(pt), cos(qt)) n(t) = (cos(pt) cos(qt), sin(pt) cos(qt), sin(qt)) These curves wrap around the rigid torus defined above p times in the longitudinal direction and q times in the meridional direction. The moving frame is then used to compute the normal curvature for a curve α in (15) and to compute the geodesic curvature in (11) The first result uses the geometry of the torus to reveal an interesting property of the curvature of torus curves. The second uses this phenomenon to describe entirely the self-linking properties of torus curves. Ultimately, the idea of the first theorem will be used in the main theorem of the paper. Theorem 2.1. Let α be a (p, q) torus curve with p, q 0 on a standard torus of revolution with radii b and 1, 0 < b < 1. Then α has points of zero curvature for only one value of b, namely b = p2. Moreover, for this value of b there are precisely q points of vanishing curvature on α, all lying on the innermost longitudinal circle of the torus. Proof. First of all, consider that by (5), the curvature will vanish if and only if the geodesic curvature and the normal curvature vanish simultaneously. Computing the geodesic curvature of α using (6) yields κ g = [ n, α, α ] ( ) dt 3 (11) ds 3 torusslv2.tex; 10/01/2003; 2:05; p.3

4 4 [ cos(qt) bq sin(qt) cos(pt) p(1 + b cos(qt)) sin(pt) = cos(pt) sin(pt) cos(qt), bq sin(qt) sin(pt) + p(1 + b cos(qt)) cos(pt), sin(qt) bq cos(qt) bq 2 cos(qt) cos(pt) + 2bqp sin(qt) sin(pt) p 2 (1 + b cos(qt)) cos(pt) ] bq 2 cos(qt) sin(pt) 2bqp sin(qt) cos(pt) p 2 (1 + b cos(qt)) sin(pt) bq 2 sin(qt) ( ) 1 (12) (p 2 (1 + b cos(qt)) 2 + q 2 b 2 )) 3 2 = (p sin(qt))(p2 (1 + b cos(qt)) + 2q 2 b 2 ). (13) (p 2 (1 + b cos(qt)) 2 + q 2 b 2 ) 3 2 Since the denominator and the second factor of the numerator of (13) are never zero, in order to have this expression vanish we must have sin(qt) = 0. The domain of α is [0, 2π] so we conclude that t must be one of the values t = kπ, k = q (14) q giving us 2q points on α to investigate. Now, in order for the curvature of α to be zero, the normal curvature of α must also be zero. One computes that κ n = II(α, α )( dt ds )2 ( ) ( ( ) ( ) = p q (1 + b cos(qt)) cos(qt) 0 p dt 2 0 b) q ds = ((1 + b cos(qt)) cos(qt))p2 + bq 2 (p 2 (1 + b cos(qt)) 2 + q 2 b 2. ) (15) The denominator of this last line is strictly positive so that all the zeroes of the normal curvature can be found by setting the numerator to zero. This yields a quadratic in cos(qt): bp 2 (cos(qt)) 2 + p 2 cos(qt) + bq 2 = 0 (16) Note that since b, p, and q are all positive, as long as cos(qt) > 0, (16) has no solutions. As a result, the values of t from (14) with k even cause the normal curvature to be strictly positive (these are the points on the outside rim of the torus). Since the geodesic curvature must vanish simultaneously with the normal curvature, we disregard these values of t. Accordingly, the only remaining solutions from (14) (where k is odd) make cos(qt) = 1 and so (16) becomes 0 = p 2 b p 2 + bq 2 = b(p 2 + q 2 ) p 2 (17) torusslv2.tex; 10/01/2003; 2:05; p.4

5 which implies that b = 5 p2 p 2 + q 2. (18) Since cos(qt) = 1 is forced by the simultaneous conditions κ g = κ n = 0, this value of b is the only one for which the given (p, q) torus curve may have points with vanishing curvature. Moreover, there are q such points for this value of b, namely {α(t) t = kπ q, k = 1, 3,..., 2q 1} (19) To see that these are indeed points of vanishing curvature for this value of b, assume (18) holds and compute κ n and κ g directly. The proof of this theorem illustrates the fact that in many instances the analysis of the curvature of a space curve may be intractable computationally, but if the curve lies on a surface whose geometry is understood then resolving the curvature of the space curve into its geodesic and normal components may yield insight. 3. Self-Linking on Rigid Tori The study of the classification of closed curves in R 3 up to homotopy and isotopy has a rich history. Homotopy classification allows deformations respecting various levels of non-degeneracy and permitting self-intersection of the curves. Isotopy adds to this the requirement that all deformations remain embeddings. Gauss defined the linking number of two curves α 1 and α 2 in space as α1 (t) α 2 (t) lk(α 1, α 2 ) = dt (20) α 1 (t) α 2 (t) Calugareanu introduced the notion of the self-linking of a space curve in [2] and Pohl showed in [7] that the Gauss integral can be extended in a well-defined manner to a single curve provided that the curvature is non-vanishing at every point of the curve. We have the following Theorem 3.1 (Pohl). The self-linking number of an embedded space curve α is the sum SL(α) = da + τds (21) where τ is the torsion of α, ds is the element of arc length and da is the pullback of the area form of S 2 under the Gauss map on the space of secants for the curve (so that the first integral is in fact the Gauss integral). torusslv2.tex; 10/01/2003; 2:05; p.5

6 6 Since the concept of a curve with everywhere non-vanishing curvature will be used several times in this paper, the following definition is made. Definition 3.2. A space curve is positive curvature if it has everywhere non-vanishing curvature. An isotopy of space curves is a positive curvature isotopy if all intermediate curves have non-vanishing curvature. Computing the self-linking invariant from these two integrals is complicated in even the best of cases so other methods have been developed for handling various examples. One such method due to Gluck and Pan [5] uses a planar projection: Lemma 3.3 (Gluck and Pan). The self linking number of a positive curvature space curve α can be computed from a planar projection as follows: SL(α) = #{positive crossings} + #{positive hills} (22) #{negative crossings} #{negative hills} (23) where a hill is defined to be a point whose image in the planar projection is an inflection of order one in the plane curve with the principal normal below the curve with respect to the direction of the projection. Inflections of order one are usually referred to as ordinary inflections. We say such a point whose image in the projection is an ordinary inflection with the principal normal above the curve with respect to the projection is a valley. A hill is positive if the rotation of the normal pushoff around the curve α agrees with the right hand rule at the hill point and negative in the other case. We remark that this method is related to the fact that the self-linking number of a space curve can be taken to be the linking number of the space curve with a curve gotten by pushing off in the principal normal direction to the space curve by a sufficiently small amount. It turns out that this geometric invariant is sufficient to classify knots with non-vanishing curvature up to isotopy. Theorem 3.4 (Gluck and Pan). Two positive curvature knots are positive curvature isotopic if and only if they have the same knot type and the same self-linking number. A partial result holds for the class of space curves with everywhere positive torsion. Theorem 3.5. If α is a (p, q) positive curvature torus curve on a torus of revolution T of variable radius b with orientation given by the torusslv2.tex; 10/01/2003; 2:05; p.6

7 parametrization above, then SL(α) = (p 1)q when 0 < b < and SL(α) = pq when p 2 < b < 1. 7 p2 Proof. From the definition of the self-linking invariant in [7] and the work of Gluck and Pan in [5] we have that the self linking number of a space curve is constant through isotopies in which all curves have nonvanishing curvature. It follows from Theorem 2.1 that the self linking number of a torus curve α is constant for values of the variable radius of the torus of revolution on which they lie in the intervals complementary to the critical value where curvature vanishes. As a result we have only to establish the value of SL(α) in each of these subintervals of [0, 2π]. To do this, begin with the following Lemma 3.6. A planar projection of a (p, q) torus curve α has ordinary p inflections only for values of b in ( 2, 1). In this case there are 2q such points. Proof. Computing the condition for ( an ordinary planar inflection of the α ) projection curve, namely that det( ) = 0, we have that α β(t) is an ordinary inflection cos(qt) = For values of p 2 p 2 b(p 2 + q 2 ) (24) < b < 1 this equation has 2q solutions and none for 0 < b < p2 (since then cos(qt) > 1). In particular, we see the 2q solutions lie as groups of two in the q subintervals ( kπ q π 2q, kπ q + π 2q ) for k = 1, 3,..., 2q 1. Consider the (p, q) torus curve for values of b lying in the interval p (0, 2 ). Since we have no ordinary planar inflections for β here we can compute SL(α) here by applying Gluck s lemma and computing the sum of the signed crossing in the planar projection, β. It is an easy fact that for a (p, q) torus curve lying on the torus of revolution that the planar projection along the axis of revolution has (p 1)q crossings. The orientation induced by the given parametrization of the curve causes these crossings to have negative sign and so SL(α) = (p 1)q for these values of b. p For the values of b in ( 2, 1), we must take into account the birth of the ordinary inflections in the planar projections. According to Lemma 3.3, negative hill type inflections contribute 1 to SL(α) whereas valleys contribute nothing. From Lemma 3.6 we see the birth of q identical pairs of inflections along the inside rim of the torus. torusslv2.tex; 10/01/2003; 2:05; p.7

8 8 Lemma 3.7. Each pair of inflections consists of a negative hill and a valley type inflection. Proof. The normal vector to the curve α is vertical with respect to the plane of projection at any inflection of the planar projection curve so determining the direction of the normal to the curve α with respect to the plane of projection at any of these planar inflections means checking the third coordinate of the principal normal. Whether this coordinate is positive (above the plane) or negative (below the plane) determines whether an inflection is a valley or a hill, respectively. In fact, since the normal vector to the curve at any point is the projection of α to the normal plane of α and this normal plane must necessarily be orthogonal to the plane of projection since the normal vector is, the sign of the third coordinate of α determines the sign of the third coordinate of the normal vector. A quick computation shows that the third coordinate of α is bq 2 sin(qt). Equation (24) implies that cos(qt) < 0 and so qt for any inflection must lie in ( kπ q π 2q, kπ q + π 2q ) for k = 1, 3,..., 2q 1. Furthermore, the two solutions to (24) in each interval lie on different sides of kπ q because of the symmetry of the arccos function and so for a given pair of solutions the one with smaller parameter t yields a point where sin(qt) > 0 and so bq 2 sin(qt) < 0 making its image in the planar projection a hill. The solution with larger t has sin(qt) < 0 and so its image is a valley. The sign of each hill is determined by the sign of the crossing formed at that point in the planar projection of the curve with the projection of its normal pushoff α ɛ = α + ɛn where N is the normal to α and ɛ > 0 is sufficiently small to make the pushoff smooth. Note that at hills α ɛ passes under α. The way in which the normal pushoff passes under the curve is reflected by the way in which the planar curvature of the planar projection changes sign as the plane curve passes through an inflection. Indeed, since the normal to the plane curve is the projection of the normal to the space curve and the normal to the plane curve is the projection of the second derivative vector of the plane curve to its normal line, the side of the curve on which the projection of the normal pushoff lies is determined by the sign of the planar curvature of the plane curve. Equivalently, the cross products of the first and second derivative vectors of the plane curve, thought of as vectors in R 3, on each side of the inflection can be compared. All that is needed to determine the sign of the crossing is a frame of reference with which to do this. The natural radial symmetry of the projection of the curve α and of the torus on which it lies yields such a frame of reference. Computing the scalar triple product of the first and second derivative vectors of torusslv2.tex; 10/01/2003; 2:05; p.8

9 the plane curve and the unit normal to the plane of projection yields 0 [α, α, 0 ] = p 3 (1+b cos(qt)) 2 +2pq 2 b 2 (sin(qt)) 2 +bpq 2 (1+b cos(qt)) cos(qt) 1 (25) whose sign determines whether or not the orientation of the first two vectors agrees with the given orientation of the xy-plane and hence the sign of the curvature of the planar projection. Let t 0 be the value of the time parameter for a hill. Equation (24) above determines the value of cos(qt 0 ) at the hill point so by continuity and the fact that the cosine function is decreasing as the curve moves toward the inner rim of the torus implies that cos(qt) > p 2 b(p 2 + q 2 ) 9 (26) when t (t 0 δ, t 0 ]. Computing (25) in this case shows that the planar curvature of the planar projection in this interval is greater than 2pq 2 b 2 (sin(qt)) 2 which is strictly positive there. Hence the planar curvature of the projected curve is positive at points immediately preceding the planar inflection. This in turn means that the projection of the normal pushoff lies radially closer to the origin than the projected curve. On the other hand, the derivative of the planar curvature of the planar projection at the planar inflection κ planar(t 0 ) = bpq3 sin(qt 0 )(5p 2 + q 2 ) p 2 + q 2 (27) which is negative since sin(qt 0 ) > 0, implying that the planar curvature is decreasing as the time parameter increases at a point where it vanishes. As a result, the planar curvature of the planar projection is strictly negative at points immediately following the planar inflection. Thus, in this region the projection of the normal pushoff lies radially farther from the origin than the projected curve. Together, these two facts show that the projection of the normal pushoff α ɛ passes under the projection of α from radially inside the curve to radially outside. This means that the crossing is in fact negative in the given orientation of the space curve α. This is locally the same as α ɛ passing from left to right under α with both curves are oriented in the same direction and so each hill in each pair of inflections is negative and contributes 1 to the self-linking number of the given torus curve. torusslv2.tex; 10/01/2003; 2:05; p.9

10 10 As a result of the lemma, (p, q) torus curves lying on tori with p variable radius b in ( 2, 1) have SL(α) = (p 1)q q = pq This completes the proof. 4. Conclusions In a letter to N. Kuiper [1], Tom Banchoff gives an argument showing that the self linking number of a (p, q) knot in the flat torus T S 3 will be pq by computing directly the linking number of the torus curve with its normal pushoff. His method of proof avoids a complication that arises when projecting knots in S 3 to a plane in order to compute the self linking number as in the proof of the main result here. For curves in R 3, we rely on the strength of Gluck and Pan s lemma (3.3). The question might then be what conditions must a planar projection from S 3 satisfy in order to capture the true self-linking number of a knot K S 3. Theorem (3.5) above also shows that the constraint of being in R 3 instead of S 3 R 4 forces the curve to flatten out on very thin tori of revolution. As a result, its self-linking invariant drops on these thin tori. References 1. Banchoff, T.: 1991, Letter to Nicolaas Kuiper. Correspondence, available at banchoff/self-linking/nk.html. 2. Călugăreanu, G.: 1961, Sur les enlacements tridimensionnels des courbes fermées. Com. Acad. R. P. Romine 11, Costa, S. I. R.: 1990, On Closed Twisted Curves. Proc. Amer. Math. Soc. 109, Fuller, E. J.: 1999, The Geometric and Topological Properties of Holonomic Knots. Ph.D. thesis, The University of Georgia. 5. Gluck, H. and L.-H. Pan: 1994, Knot theory in the presence of curvature. preprint. 6. Little, J. A.: 1978, Space Curves with Positive Torsion. Ann. Mat. Pura Appl. (4) 116, Pohl, W. F.: 1968, The self-linking number of a closed space curve. J. Math. Mech. 17, Romero Fuster, M. C.: 1988, Convexly generic curves in R 3. Geom. Dedicata 28(1), Spivak, M.: 1979, A Comprehensive Introduction to Differential Geometry: Volume II, Second Edition. Berkeley, CA: Publish or Perish. torusslv2.tex; 10/01/2003; 2:05; p.10

11 10. Struik, D. J.: 1961, Lectures on Classical Differential Geometry. New York: Dover Publications, Inc. 11 torusslv2.tex; 10/01/2003; 2:05; p.11