The Point-Split Method and the Linking Number of Space Curves
Timmy Forsberg∗†
Uppsala University Division of Theoretical Physics
Degree Project C 15 hp
Aug 06 2014
Abstract
This is a report on research done in the field of mathematical physics. It is an inves- tigation of the concept of the linking number between two simple and closed spatial curves. The linking number is a topological invariant with scientific applications ranging from DNA biology to Topological Quantum Field Theory. Our aim is to study C˘alug˘areanu’s theorem, also called White’s formula, which relates the linking number to the concepts of twist and writhe. We are interested in the limit of the two curves as they approach each other. To regulate this, we introduce a regulariza- tion method that utilizes a point-split. Further we explore if the result is dependent on how the regularization is introduced. Therefor we inflict an asymmetry in the regularization, with a parameter a in the point-split intervals, to check whether the result becomes dependent on a or not. We find that the result is independent of the parameter a.
∗Supervisor: Antti Niemi
†Subject reader: Joseph Minahan
Contents
1 Introduction 1
2 The Linking Number 2
3 Lk = Tw + Wr and Symmetric Regularization 4
3.1 The Writhing Number and Symmetric Regularization. . . 5 3.2 The Twisting Number and Symmetric Regularization. . . 6
4 Lk = Tw + Wr and Asymmetric Regularization 7
4.1 Writhing Number and Asymmetric Regularization . . . 7 4.2 Twisting Number and Asymmetric Regularization . . . 8
5 Conclusion 9
References 9
1 Introduction
The begining of Knot theory is in a brief note made in 1833, where Carl Friedrich Gauss introduces a mathematical formula that computes the linking number of two space curves[0]. The linking number, using Gauss own (translated) words, “...count the intertwinings of two closed or endless curves.”. Which means it measures the extent to which the two curves have been knotted about each other. Subsequently, Peter Tait, Sir William Thomson (Lord Kelvin) and James Clerk Maxwell [1]; started developing Knot Theory with physical applications in mind. Maxwell successfully applied it to electromagnetic theory. Lord Kelvin suspected that atoms could be knotted vortices in the aether and that this could help explain the stability of matter and why light is absorbed and emitted as quanta. Tait who was also keen on that idea, worked on more mathematical aspects, creating conjectures and initiated a tabulation process of the simplest knots. In 1887 the Michelson-Morley experiment results in a rapid decline of knot theory in physics and is soon dissolved. But the mathematicians interest in knots and their properties was continued. They developed knot theory into a subclass in the study of manifolds and topology.[1] [2]
In the late 20th century, knot theory returned into physics with the advancement of Topological Quantum Field Theory (QTFT) and Loop Quantum Gravity. Around the same time, knot theory had also found its use in DNA biology and chemistry, in connection to circular DNA and sterioisomers.[1] [3]
In this research project we derive C˘alug˘areanu’s theorem, which describes the linking number (Lk) of two space curves in terms of writhe (W r) and twist (T w). Whenever a link of two knots can be estimated as the edges of a ribbon, C˘alug˘areanu’s theorem says
that:
Lk = W r + T w
We derive this relation in the general way by analyzing what happens as two separate curves approach each other, until they completely overlap. It is possible that an anomaly exists, so that the result might depend on how the two curves approach each other. To study this, we introduce a regularization scheme called the point-split method.
2 The Linking Number
Consider a string-like spatial curve, simple and closed, i.e. its ends is joined together.
This object is what mathematicians call a knot : An embedding of the circle in a three dimensional space. A coordinate system can be established at each point of the curve with the use the Frenet-Serret formulas:
d ˆT dt =κ ˆT d ˆN
dt = − κ ˆT + τ ˆB d ˆB
dt = − τ ˆN
building up the Frenet frame. ˆT, ˆN and ˆB are the unit tangent, unit normal and unit binormal vectors. They are perpendicular to each other and form the coordinate system at each point of a curve C. τ and κ is the torsion and curvature. As long as κ 6= 0, the curve can be uniquely determined its curvature κ(t) and its torsion τ (t); note that the curve is parametrized by the arc length (t). The Frenet frame satisfy the following equations:
• T(t) × T(t) = 0
• ˆT(t) · ˆT(t) = 1
• dT(t)dt · T(t) = 0
• N(t) · T(t) = 0
Now, consider two non-intersecting knots that are tangled together, perhaps in the same way as Figure 1. Such a system is called a link.
As shown in Figure 1, the link is two embeddings of a circle in R3. The two knots cannot intersect, because otherwise the system is ill-defined. The Linking Number or Lk, as it is sometimes shortened, is an integer invariant measure on how linked a system
Figure 1: This is a link. The knots do not actually have a width, it is merely an artistic feature.
of two knots is. The Linking Number can be calculated in many different ways, although Gauss Linking Integral is probably the most well known:
Lk = 1 4π
I
C0
dt0 I
C
dt r0(t0) − r(t)
|r0(t0) − r(t)|3 · dr
0(t0)
dt0 ×dr(t) dt
(1)
where r and r0 is the vectors from the origin to a point on each of the curves C and C0.[4]
Gauss Linking Integral can be derived by calculating the flux from a solenoidal field across an oriented surface, where the field only has values along one of the knots, and the surface is spanned by the other knot. Another way to do it is to calculate Ampere’s law with Bio-Savart’s law as the magnetic field B, with constant flow of current in a closed curve C.[5] (µ0I is put to one because the linking number only attends to topological properties.)
B(x0) = 1 4π
I
C
dl × r
|r|3
= 1 4π
I
C
dx × (x0− x)
|x0− x|3
⇒ Lk = I
C0
B(x0) · dx0
= 1 4π
I
C0
I
C
dx × (x0− x)
|x0− x|3 · dx0
= 1 4π
I
C0
I
C
dx0× dx · (x0− x)
|x0− x|3 (We’ll later use r and r0 instead of x and x0.)
Figure 2: An illustration of Twisting and Writhing
Figure 2 illuminates the concepts of twisting and writhing. Twisting arise from the individualistic behavior of two nearby knots, whereas writhing is more general. One may rewrite Lk by imagining that the link’s two curves (C and C0) lie on the edges of an orientable ribbon. C˘alug˘areanu’s theorem (also called White’s formula) then says that:
Lk(C, C0) = T w(C, C0) + W r(C) (2) where T w is The Twisting Number and W r is The Writhing Number.[6] Writhing is what generally happens to a ribbon when it’s strongly twisted and there is a significant decrease in string tension (locally or globally). C˘alug˘areanu’s theorem made its success in DNA research, and it also led to advances within TGFT (Topological Quantum Field Theory).[7][8]
Our objective is to derive Eq. (2), in the limit of the two curves as they approach each other, by using a symmetrical point-split regularization. We also want to determine whether the result can depend on how the two curves approach each other. Which is why our main goal is to preform an asymmetrical point-split regularization to the two curves.
If it is found that the result do depend on the regularization, i.e. the way we introduce the point-split, there is an anomaly, which could have implications in mathematical physics. We start by re-deriving the original C˘alug˘areanu’s theorem which utilizes the symmetrical point-split, and after that we proceed with the asymmetrical generalization.
3 Lk = Tw + Wr and Symmetric Regularization
We start from Gauss Linking integral Eq. (1), in the context of a ribbon we assume two non-intersecting, non-self-intersecting and closed curves that go around each other at a
small distance . The following relation can then be made for any ribbon.[9]
r0(t0) = r(t0) + n(t0) (3) From the Frenet frame note that in Eq. (3), n(t0) is the unit normal at r(t0) that points through r0(t0). Also T(t0) is the tangent vector as shown in Figure 3.
Also note that:
dr0(t0)
dt0 = r(t0)
dt0 + dn(t0)
dt0 = T(t0) + dn(t0)
dt0 (4)
Figure 3: Imagery of the vectors and their notations.
Onwards, Eq. (1), Eq. (3) and Eq. (4) gives us that:
1 4π
I
C0
dt0 I
C
dt r(t0) + n(t0) − r(t)
|r(t0) + n(t0) − r(t)|3 ·
(T(t0) + dn(t0)
dt0 ) × T(t)
(5) When is much smaller than δ, our regularization is fine. Eq. (5) can be written as the sum of a singular part and a non-singular part separated at t = t0.
3.1 The Writhing Number and Symmetric Regularization.
The non-singular part of Eq (5) is:
1 4π
I
C0
dt0 lim
δ→0+
Z t0−δ 0
dt + Z L
t0+δ
dt
r(t0) + n(t0) − r(t)
|r(t0) + n(t0) − r(t)|3 ·
T(t0) + dn(t0)
dt0 × T(t)
(6) Where 0 and L denotes the arbitrary beginning and endpoint of the curve C. Notice that in Eq. (6) there exists no singular points, and hence is abundant. As the limit of
δ is taken to 0 with << δ, we notice that the two curves C and C0 now must lie on top of each other. This is must be the Writhing Number W r:
W r = 1 4π
I
C
dt0 I
C
dt r(t0) − r(t)
|r(t0) − r(t)|3 ·
T(t0) × T(t)
(7)
3.2 The Twisting Number and Symmetric Regularization.
The singular part of Eq (5) is:
1 4π
I
C0
dt0 Z t0+δ
t0−δ
dt r(t0) + n(t0) − r(t)
|r(t0) + n(t0) − r(t)|3 ·
T(t0) + dn(t0)
dt0 × T(t)
(8) Let’s start by approximating r(t) around t0to linear terms (higher terms would eventually yield zero anyway):
(r(t) ≈ r(t0) + (t − t0)T(t0)
T(t) ≈ T(t0) + (t − t0)dT(tdt0) (9) ( (t − t0)dT(tdt0) is a subleading term and can be removed, but we have chosen to keep it for expository reasons)
We place Eq. (9) into Eq. (8) and use that T(t0) × T(t0) = 0 together with the additive property of the cross product, we get:
= 1 4π
I
C0
dt0 Z t0+δ
t0−δ
dt 1
|n(t0) − (t − t0)T(t0)|3
+ n(t0) ·dn(t0)
dt0 × T(t0)+
+ n(t0) ·T(t0) × (t − t0)dT(t0) dt +
+ n(t0) − (t − t0)T(t0) · dn(t0)
dt0 × (t − t0)dT(t0) dt
Simplifying the denominator using that n(t0) · T(t0) = 0 and n(t0) · n(t0) = 1 gives:
= 1 4π
I
C0
dt0 Z t0+δ
t0−δ
dt 1
p(t − t0)2T2(t0) + 23
+ n(t0) · [dn(t0)
dt0 × T(t0)]+
+ n(t0) · [T(t0) × (t − t0)dT(t0) dt ]+
+ n(t0) · [dn(t0)
dt0 × (t − t0)dT(t0) dt ]−
− (t − t0)T(t0) · [dn(t0)
dt0 × (t − t0)dT(t0) dt ]
Taking the integrals:
1 4π
I
C0
dt0n(t0) · [dn(t0)
dt0 × T(t0)] 2(t − t0)
2p2+ (t − t0)2T(t0)2
t=t0+δ t=t0−δ
+
+ 1 4π
I
C0
dt0n(t0) · [T(t0) ×dT(t0)
dt ]
T(t0)2p2+ (t − t0)2T2(t0)
t=t0+δ t=t0−δ
+
+ 1 4π
I
C0
dt0n(t0) · [dn(t0)
dt0 ×dT(t0)
dt ] 2
T(t0)2p2+ (t − t0)2T2(t0)
t=t0+δ t=t0−δ
−
− 1 4π
I
C0
dt0T(t0) · [dn(t0)
dt0 ×dT(t0) dt ]
Z t=t0+δ t=t0−δ
dt (t − t0)2
2+ (t − t0)2T2(t0)3/2
(10)
The unfinished integral isn’t written out because of its length. But taking the limit of δ with << δ only leaves the top term in Eq. (10), which then is the twisting number.
T w = 1 4π
I
C0
dt0 (t − t0)n(t0)
|(t − t0)||T(t0)|· dn(t0)
dt0 × T(t0)
t=t0+δ
t=t0−δ
= 1 2π
I
C0
dt0 n(t0)
|T(t0)|· dn(t0)
dt0 × T(t0)
(11)
Note that Eq. (11) is independent of the magnitude of T(t0).
4 Lk = Tw + Wr and Asymmetric Regularization
In the previous section we introduced a point-split with a symmetric interval [−δ, δ].
In this section we generalize it to the case with an asymmetric interval [−δ1, δ2] where δ2 = aδ1 and a is a parameter. Following the computations of the previous section up to Eq. (5), because of the similarities in the methods. We generalize this equation, embedding its singular contribution along the curve C with the interval of [−δ1, aδ1].
See Figure 4.
Note that, we could let the limit of δ1 be taken with << aδ1 and << δ1. As that directly gives us the same result as in the previous section. We will instead look at what values the parameter a can take, while still leaving us with C˘alug˘areanu’s theorem.
4.1 Writhing Number and Asymmetric Regularization The non-singular part of Eq. (5) is:
1 4π
I
C0
dt0 lim
δ1→0+
Z t0−δ1
0
dt+
Z L t0+aδ1
dt
r(t0) + n(t0) − r(t)
|r(t0) + n(t0) − r(t)|3·
T(t0)+dn(t0)
dt0 ×T(t)
(12) Notice that in Eq. (12), on the left hand side of the point t = t0, [t0− δ1, t0], there exists no singular points that we might evaluate in, so that interval will contract without
Figure 4: A way of visualizing the problem ahead.
any problems. However on the right hand side of the point t = t0, [t0, t0+ aδ1], we need to at least evoke
a ≥ 1
in order to get from Eq. (12) to Eq. (7), since this implies that << aδ1. 4.2 Twisting Number and Asymmetric Regularization The singular part of Eq. (5) is:
1 4π
I
C0
dt0
Z t0+aδ1
t0−δ1
dt r(t0) + n(t0) − r(t)
|r(t0) + n(t0) − r(t)|3 ·
T(t0) + dn(t0)
dt0 × T(t)
(13) By the same reasoning and procedure as in the previous section, Eq. (13) becomes Eq.
(14):
1 4π
I
C0
dt0n(t0) · [dn(t0)
dt0 × T(t0)] 2(t − t0)
2p2+ (t − t0)2T(t0)2
t=t0+aδ1
t=t0−δ1
+
+ 1 4π
I
C0
dt0n(t0) · [T(t0) × dT(t0)
dt ]
T(t0)2p2+ (t − t0)2T2(t0)
t=t0+aδ1
t=t0−δ1
+
+ 1 4π
I
C0
dt0n(t0) · [dn(t0)
dt0 ×dT(t0)
dt ] 2
T(t0)2p2+ (t − t0)2T2(t0)
t=t0+aδ1
t=t0−δ1
−
− 1 4π
I
C0
dt0T(t0) · [dn(t0)
dt0 ×dT(t0) dt ]
Z t=t0+aδ1
t=t0−δ1
dt (t − t0)2
2+ (t − t0)2T2(t0)3/2
(14)
Observe that for a ≥ 1, all singular points are included because the limit of δ1 is taken towards zero with << δ1 ≤ aδ1, so:
T w = 1 4π
I
C0
dt0 (t − t0)n(t0)
|(t − t0)||T(t0)|· dn(t0)
dt0 × T(t0)
t=t0+aδ1
t=t0−δ1
=
( a
|a|+ 1) δ1
|δ1|
1 4π
I
C0
dt0 n(t0)
|T(t0)|· dn(t0)
dt0 × T(t0)
= 1 2π
I
C0
dt0 n(t0)
|T(t0)|· dn(t0)
dt0 × T(t0)
for a ≥ 1
(15)
5 Conclusion
We started with the definition of Gauss linking integral, introduced a point-split regu- larization, and from that derived C˘alug˘areanu’s theorem:
Lk =
= W r + T w
= 1 4π
I
Caxis
dt0 I
Caxis
dt r(t) − r(t0)
|r(t) − r(t0)|3 ·T(t) × T(t0) + 1 2π
I
C0
dt0 n(t0)
|T(t0)|·T(t0) ×dn(t0) dt0
(16) We got the same results intependently from the regularization procedures, where Eq.
(16) was the result for both the symmetric and asymmetric point-split intervals. The parameter a that we used to introduce the asymmetric point-split, had to be greater or equal to one, but did not appear in the final result; hence there is no anomaly.
References
[1] Ricca, L.R., Nipoti, B. (2011) Gauss Linking Number Revisited, J. Knot Theory Ramifications 20, 1325.
[2] Kauffman, L. H. (2004) Review of ”Knots: Mathematics with a Twist (by Alexei Sossinsky)” The Amer. Math. Monthly. 111, no. 9, 830-838.
[3] Sossinsky A. (2002) Knots: Mathematics with a Twist. Harvard University Press, Cambridge
[4] Adams, C. Collin, The Knot Book.
[5] Kamien, R. D. The geometry of soft materials: a primer.
[6] De Zela, F. (2004) Linking Maxwell, Helmholtz and Gauss through the Linking Integral. arXiv:physics/0406037v1
[7] C˘alug˘areanu, G. (1961) Sur les classes d’isotopie des noeuds tridimensionnels et leurs invariants. Czech. Math. J. 11, 588-625.
[8] Kauffman, L. H. (2001) Knots and physics (3rd Edition), part II, Chap. 15.
[9] Witten, E. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121 (1989), no. 3, 351-399.
[10] White, J. H. Self-linking and the Gauss integral in higher dimensions. Amer.
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