Handbook of Knot Theory


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For knots to be truly useful, however, mathematicians searched for some precise way of proving that what appeared to be different knots such as the trefoil knot and the figure eight knot were really different—they couldn't be transformed one into the other by some simple manipulation.

Unreasonable Effectiveness of Knot Theory

Towards the end of the nineteenth century, the Scottish mathematician Peter Guthrie Tait and the University of Nebraska professor Charles Newton Little published complete tables of knots with up to ten crossings. Unfortunately, by the time that this heroic effort was completed, Kelvin's theory had already been totally discarded as a model for atomic structure. Still, even without any other application in sight, the mathematical interest in knot theory continued at that point for its own sake. The only difference was that, as the British mathematician Sir Michael Atiyah has put it, "the study of knots became an esoteric branch of pure mathematics.

Figure 2. Deforming a knot.

One of the main goals of knot theory has always been to identify properties that truly distinguish knots—to find what are known as knot invariants. A knot invariant acts very much like a "fingerprint" of the knot; it does not change by superficial deformations of the knot for example, of the type demonstrated in Figure 2.

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At first blush, you may think that the minimum number of crossings in a knot could serve as such an invariant. After all, no matter how hard you try, you will never be able to reduce the number of crossings of the trefoil knot see Figure 1b to fewer than three. However, the minimum number of crossings is actually not a very useful invariant. As Figure 1 demonstrates, there are three different knots with six crossings and no fewer than seven different knots with seven crossings. Figure 3. A knot and its mirror image. Two major breakthroughs in knot theory occurred in and in In , the American mathematician James Waddell Alexander discovered an algebraic expression known as the Alexander polynomial that uses the arrangement of crossings to label the knot.

Consequently, while it was certainly very useful, the Alexander polynomial was still not perfect for classifying knots. Decades of work in the theory of knots finally produced the second breakthrough in The New Zealander-American mathematician Vaughan Jones detected an unexpected relation between knots and another abstract branch of mathematics known as von Neumann algebras. This led to the discovery of a more sensitive invariant than the Alexander polynomial, which became known as the Jones polynomial.

The Jones polynomial distinguishes, for instance, even between knots and their mirror images see Figure 3 , for which the Alexander polynomials were identical. More importantly, the flurry of activity that ensued following Jones's discovery suddenly connected a bewildering variety of areas in mathematics and physics, and penetrated even into string theory—the current most promising attempt to reconcile general relativity with quantum mechanics.

In particular, string theorists Hirosi Ooguri and Cumrun Vafa discovered that the number of complex topological structures that are formed when many strings interact is related to the Jones polynomial. Furthermore, the leading string theorist Edward Witten demonstrated that the Jones polynomial affords new insights in one of the most fundamental areas of research in modern physics, known as quantum field theory.

The lesson from this very brief history of knot theory is remarkable. First, it was the active effectiveness of mathematics that came into play. Physicists needed a model for the atom, and when knots appeared to provide the appropriate tool, a mathematical theory of knots took off. When a better mathematical model in the form of the Bohr atom was discovered, mathematicians did not abandon knot theory. Driven only by their curiosity, they continued to explore the properties of knots for many decades. The mere possibility of understanding the properties of knots and the principles that govern their classification was seen by most mathematicians as exquisitely beautiful and essentially irresistible.

However, then came the surprising passive effectiveness of mathematics. Unexpectedly, the Jones polynomial and knot theory in general turned out to have wide-ranging applications in string theory. Knots leading the way, from the atom to pure maths and back to physical matter.

What makes this story even more striking is the following fact.

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Recall that Thomson started to study knots because he was searching for a theory of atoms, then considered to be the most basic constituents of matter. By a remarkably circular twist of history, knots are now found to provide answers in string theory, our present-day best effort to understand the constituents of matter!

So knot theory emerged from an attempt to explain physical reality, then it wandered into the abstract realm of pure mathematics—only to eventually return to its ancestral origin. Isn't this absolutely amazing? Wigner, Eugene P. Mario Livio's book, Is God a Mathematician?

Why Knot? An Introduction to Knot Theory

It follows the lives and thoughts of some of the greatest mathematicians in history, and attempts to explain the "unreasonable effectiveness" of mathematics. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Citing Literature. Volume 18 , Issue 3 August Pages Related Information. Close Figure Viewer. Browse All Figures Return to Figure.

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  • Knot Theory | profhugodegaris.
  • On diagrammatic bounds of knot volumes and spectral invariants!
  • 1st Edition.
Handbook of Knot Theory Handbook of Knot Theory
Handbook of Knot Theory Handbook of Knot Theory
Handbook of Knot Theory Handbook of Knot Theory
Handbook of Knot Theory Handbook of Knot Theory
Handbook of Knot Theory Handbook of Knot Theory
Handbook of Knot Theory Handbook of Knot Theory

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