You should be able to find the rule for the number of strands in the sequence of torus knots and links, and to explain it. It is harder to find out which are 3-colourable, and to determine the crossing number. The crossing number and the unknotting number of torus knots have been found, as you can see from our page on Torus Knots 2.
W. Ashley writes in his `Book of Knots':
`But there is always another car ahead in a possible parking place. Here is a Mr. Klein asserting that knots cannot be tied in four dimensions.'
To see why knotted string can be untied in four dimensions, we work by analogy. The jump from 2 dimensions to 3 dimensions can be illustrated by an ant moving on a table. He can only get past a long obstacle by going over it, that is, into the third dimension.
To untie a piece of knotted string, you move one piece of string against another and then past it by moving into the 4th dimension (in your imagination!).
In science and mathematics there is always another question in any parked answering place. If knotted string can be untied in 4 dimensions, what then can be tied?
The answer to this question is a knotted balloon, or sphere.
Ashley also writes:
`From the earliest times to the present day, the joy incident upon occasional discovery has ever been considered sufficient reward in itself for any human effort or sacrifice.'
So mathematics tries to understand the structure and patterns of objects we cannot make, and can only perceive as through a glass, darkly. For this the tools of study and certainty are not experiment but the development of language, concepts and logic to describe the previously indescribable, to `give to airy nothing, a local habitation and a name' (A Midsummer Night's Dream, W M Shakespeare).
| Links and Families |
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
This material may be used freely for educational, artistic and scientific purposes, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers.