Seifert Surfaces for Knots and Links.

We are all familiar with knots. Knots keep our boats securely moored and enable us to sail across the oceans. They also reduce the cables and wires behind our computers to a tangled mess. Many fabrics are just complicated knots of fibre and we know how they can unravel.

From left to right: Figure-of-8 knot 4_1, knot 6_3 , knot 7_1, and knot 8_5 [Drawn with SeifertView (image from Van Wijk, 2006)].

If the ends of a rope are free, any knot can be untied, but if the ends are spliced together it cannot be undone. To a mathematician, a knot is like a rope with the ends joined. It is intrinsically equivalent (or homeomorphic) to a circle. But extrinsically, knots differ in the way they are embedded in three dimensional space.

Knot theory is a subfield of topology. A central problem of the theory is to classify all possible knots and to determine whether or not two knots are equivalent. Equivalence means that one knot can be transformed into the other by a continuous distortion without breaking or passing through itself. Technically, this transformation is called an ambient isotopy.

In practice, knots are often distinguished by means of knot invariants. An invariant is a quantity that remains unchanged no matter how the knot is distorted. If its value differs for two knots, they cannot be equivalent. Many invariants are known, but no single one is sufficient to distinguish between all inequivalent knots.

Millions of knots and links have been tabulated since the emergence of knot theory in the 1800s. The figure below shows a few elementary examples, a simple circular loop called the unknot, and a trefoil knot together with its mirror image.

Simple knots. On the left is the unknot. Centre and right are a trefoil knot and its mirror image [Wikimedia Commons].

All of these knots are distinct: none of them can be transformed into another one. There are knot invariants, such as the Jones polynomial, that take different forms for each of these three knots.

If two or more knots are intertwined, we have a link. A link has multiple components, each being a knot. A few basic links are shown below. The simplest is composed of two separate unknots and is called the unlink. The centre image is the Hopf link, the simplest possible non-trivial link. On the right are the Borromean rings, remarkable in that no two of the three components are linked but the trio of unknots is inextricably intertwined. Amazing!

Kelvin’s Theory of Vortex Atoms

In 1867 the physicist William Thompson, later Lord Kelvin, developed an atomic theory using knots. He speculated that atoms were vortices in the aether, different kinds of atoms being knotted in different ways. Each kind of atom would have a different kind of knot. Atoms could combine like tangled smoke rings.

This theory led nowhere in physics, but it inspired Kelvin’s colleague Peter Guthrie Tait to create an extensive catalogue of knots, analogous to the periodic table of the elements. This triggered the mathematical study of knots as a sub-branch of topology. In some ways, Kelvin’s idea was reminiscent of current efforts to model the fundamental forces of nature using string theory.

Seifert surfaces

The unknot is similar to a circle, and we can regard it as the boundary of a disk-like surface. Surprisingly, we can find surfaces with knotted boundaries. In fact, every knot or link can be represented as the boundary of a surface: given any knot, we can construct an orientable (two-sided) surface having the knot as its boundary. In 1934 the German mathematician Herbert Seifert (1907-1996) devised an algorithm for constructing such a surface for any knot or link.

Seifert surfaces can be used to study the properties of knots and links. Many knot invariants can be calculated using these surfaces. Seifert defined the genus of a knot via the genus of its Seifert surface. Genus is a knot invariant.

Any compact, connected oriented surface is the Seifert surface of its boundary link. A disk has the unknot as its boundary. The Möbius strip also has this boundary but, since it is not orientable, it is not a Seifert surface.

SerfertView

The Seifertview program of Jack van Wijk is a freely available software package to compute and visualise Seifert surfaces. The trefoil knot and its Seifert surface, drawn using this package, are shown in the following figure.

The trefoil knot (left) is a simple overhand knot with the ends joined. Right:  Seifert surface, a smooth compact connected and non-self-intersecting surface, whose only boundary component is the trefoil knot.

The SeifertView package enables knots and surfaces to be viewed from any angle. Another view of the trefoil knot and its Seifert surface is shown below (left). The Seifert surface with the boundary knot removed is shown on the right. It is like two disks joined by three bands, each with a half-twist.

Another view of the trefoil knot and its Seifert surface, drawn with SeifertView. Right: the Seifert surface with its boundary knot removed.

Summary

Physical science often leads to the development of new areas of mathematics. Mathematicians may then carry the study of these areas far beyond the immediate needs of physics. This research may prove useful at a later stage in new problems in physics. This was the case with knot theory. Inspired by Kelvins vortex atoms, it took on a life of its own, developed by mathematicians but ignored by physicists until recent decades. It is now important for the study of DNA entanglement and in unified theories of fundamental physics.

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