On Knots and Links

The picture below is of a sculpture piece called Intuition, which stands in front of the Isaac Newton Institute (INI) in Cambridge. It is in the form of the Borromean Rings, a set of three interlocked rings, no two of which encircle each other.

intuition-dscf0061-lores

“Intuition”. A sculpture piece in front of the Isaac Newton Institute [Photograph courtesy of S J Wilkinson].

Knot Theory

Knot theory is an active research area today. In addition to its intrinsic mathematical interest, it has much value in theoretical physics and in biochemistry. In recent decades, scientists have been using knot theory to understand the properties of DNA and other large molecules. Chemical species with mechanical rather than electronic bonds between large molecules are being developed, and these “nano-machines” have huge potential for future applications.

Knot theory had its origins in the work of three Scottish physicists, William Thompson (Kelvin), Peter Guthrie Tait and James Clark Maxwell. In 1885, Tait published a table of mathematical knots with up to ten crossings. Showing that two knots are equivalent is usually easy: we find a way of continuously deforming one into the other. To show that two knots are not equivalent, we must prove that no such deformation is possible; this is usually a much more difficult task.

trefoil-knots-3

Trefoil knot from three different viewpoints

A (mathematical) knot is a set in R3 homeomorphic to the unit circle S1. The knot can be mapped continuously onto the unit circle, but normally cannot be continuously deformed onto it without self-intersection. All knots that can be deformed onto S1 are equivalent and are known collectively as the unknot. More generally, the mapping between equivalent knots K1 and K2 is an “ambient isotopy”, a continuous map on R3 that transforms K1 to K2 without self-intersection.

Links

hopf-linkA link comprises two or more intertwined knots. Two separated circles form the unlink. The simplest nontrivial link is the Hopf Link, in which each of two components loops once around the other. The Linking number for each is 1.

Each pair of components of the Borromean Rings has linking number zero: that is, each pair forms an unlink. But the three components together are inextricably intertwined and cannot be separated without breaking or cutting one of the components.

borromeo3dbothviews

Borromean Rings: Top view (left) and oblique view (right).

borromeo3d

imu-logo

IMU logo

The Borromean Rings cannot be realized in the plane. From above, each component looks circular, but they must be distortions of circles to avoid intersecting each other. A simple realization is the combination of three ellipses oriented along the primary axes.

In 2006, the International Mathematical Union (IMU) introduced a new logo based on the three-dimensional version of the Borromean rings. The logo represents “the interconnectedness of the various fields of mathematics, and of the mathematical community around the world”!

The Whitehead Link

Another interesting link is the Whitehead Link, with two components. In the usual diagram, the two components appear different in character:

whitehead-link

The Whitehead Link

However, this is a matter of perspective. Viewing them from another angle, we see that each component is the unknot:

whitehead-pieces

Components of the Whitehead Link.

Knots and links can be formed from curves on the surface of a torus. An example of a two-component link called the (2,8) torus link is shown here. For more information on the linking number, see the Wikipedia article referenced below.

3d-link

3D Link – the (2,8) torus link. [Wikimedia Commons]

Trailer: I hope to consider knots, links and nano-robotics in a future post.

Sources

Peter Cromwell, 2004: Knots and Links. Cambridge Univ. Press. ISBN: 978-05215-4831-1.

The Knot Atlas:  http://katlas.org/wiki/Main_Page

Wikipedia: Linking Number

*        *        *

The collection tm-cover-detail-thumbThat’s Maths, with 100 articles, has just been published by Gill Books. Available from


Last 50 Posts

Categories