Undergraduate Research Opportunity Programme in Science
Linking Number & Potential Theory
Lee C.H.1 and Wong Y.L.2
Department of Mathematics, National University of Singapore
2 Science Drive 2, Singapore 117543.
Abstract:
This paper aims to show the proof of the theorem that the linking number between
two links k, k’ is given by
1
4

k
k'
(k  k ')  dk  dk '
k -k'
3
. This result is not proven explicitly in
most literature.
The paper begins with a general introduction to linking numbers. The
combinatorial definition of linking numbers is given and a general exposition of
combinatorial knot theory is given (Reidemeister moves).
A Reidemeister move comprises changing a portion of a link diagram in one of
the following ways:
1.
A single strand may be twisted, adding a crossing of the strand with itself,
or a crossing of a strand with itself can be removed by untwisting the strand.
2.
When two strands run parallel one may be pushed under the other creating
two new over-crossings, or when a strand has two consecutive over-crossings with
another single strand both strands can be straightened to remove both over-crossings and
allow them to run parallel.
1
2
Student
Doctor
3.
Given three strands labelled A, B and C so that A passes below B and C, B
passes between A and C, and C passes above A and B, the strand A may be moved to
either side of the crossing of B and C.
In pictures:
In connection to this, Ampère’s law is derived by showing that the field generated
around a loop is irrotational. This requires the use of vector calculus.
Consider a circular loop L1. On any given point x not on L1, and a point r on the
curve, the distance between x and r is given by x r and the scalar field generated
is f ( x)  
1
x r

1
( x1  r1 ) 2  ( x2  r2 ) 2  ( x3  r3 ) 2
. To show that F( x) is irrotational,
it is sufficient to show that the curl of F( x) is equal to 0 . At the same time, the physics
method of deriving Ampère’s law is also shown, to show the close link between
mathematics and theoretical physics.
The use of divergence theorem allows us to use the result from Ampère’s law to
show the Biôt-Savart law. The Divergence Theorem can be used to evaluate the above
double integral. Divergence Theorem relates the vector surface integral over a closed
surface to a triple integral over the solid region enclosed by the surface. The theorem is
given as such: Let D be a bounded solid region in 3 whose boundary D consists of
finitely many piecewise smooth, closed orientable surfaces, each of which is oriented by
unit normals n that point away from D. Let F be a vector field whose domain includes
D. Then

D
F  dS    F dV   F  n dS .
D
S
Ampère’s law was obtained from a single loop but Biôt-Savart law is from two
loops linked together (Hopf link). Here, a number will be found which we call the
Gauss’ linking number. Once again, the physics method of deriving Biôt-Savart law is
shown, for further comparison.
Finally, topology is called into action to reduce any 2 component link into a series
of simple Hopf links, and Stokes’s theorem is used to justify that these Hopf links will
generate the Gaussian linking number of the link. Seifert surfaces will be required to
show that the links can be reduced to simple Hopf link. Choose a diagram for the link in
the xy-plane in 3 . In a small neighbourhood of each crossing, make the following local
change to the diagram: delete the crossing and reconnect the loose ends in the only way
compatible with the orientation. When this has been done, the diagram becomes a set of
disjoint simple loops in the plane – it is a diagram with no crossings. These loops are
called Seifert circles.
The Gaussian linking number of the link will be of the same value as the linking
number of the link obtained from the combinatorial definition.
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Rolfsen, Dale. Knots and Links. Wilmington: Perish Or Die, Inc., 1976
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