Knot theory

Knot theory is a branch of topology inspired by observations, as the name suggests, of knots. But progress in the field does not depend exclusively on experiments with twine. Knot theory concerns itself with abstract properties of theoretical knots — the spatial arrangements that in principle could be assumed by a loop of string.

An introduction to knot theory

Creating a knot is easy. Begin with a one-dimensional line segment, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. One of the biggest unresolved problems in knot theory is to give a method to decide in every case whether two such embeddings are different or the same.

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The unknot, and a knot equivalent to it

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Before we can do this, we must decide what it means for embeddings to be "the same". We consider two embeddings of a loop to be the same if we can get from one to the other by a series of slides and distortions of the string which do not tear it, and do not pass one segment of string through another. If no such sequence of moves exists, the embeddings are different knots.

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A useful way to visualise knots and the allowed moves on them is to project the knot onto a plane - think of the knot casting a shadow on the wall. Now we can draw and manipulate pictures, instead of having to think in 3D. However, there is one more thing we must do - at each crossing we must indicate which section is "over" and which is "under". This is to prevent us from pushing one piece of string through another, which is against the rules. To avoid ambiguity, we must avoid having three arcs cross at the same crossing and also having two arcs meet without actually crossing. When this is the case, we say that the knot is in general position with respect to the plane. Fortunately a small perturbation in either the original knot or the position of the plane is all that is needed to ensure this.

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Reidemeister moves

In 1927, working with this diagrammatic form of knots, J.W. Alexander and G.B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown right. These operations, now called the Reidemeister moves, are:

Related Topics:
J.W. Alexander - G.B. Briggs - Kurt Reidemeister

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• Twist and untwist in either direction.
  
• Move one loop completely over another.
  
• Move a string completely over or under a crossing.
  

Knot invariants can be defined by demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves. Many important invariants can be defined in this way, including the Jones polynomial. Older examples of knot invariants include the fundamental group and the Alexander polynomial.

Related Topics:
Knot invariants - Jones polynomial - Fundamental group - Alexander polynomial

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Higher dimensions

You can unknot any circle in four dimensions. There are two steps to this. First, "push" the circle into a 3-dimensional subspace. This is the hard, technical part which we will skip. Now imagine temperature to be a fourth dimension to the 3-dimensional space. Then you could make one section of a line cross through the other by simply warming it with your fingers.

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In general piecewise-linear n-spheres form knots only in n+ 2 space, although one can have smoothly knotted n-spheres in n+ 3 space.

Related Topics:
N-sphere

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Adding knots

Two knots can be added by cutting both knots and joining the pairs of ends. Knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the n-spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions.

Related Topics:
Monoid - Prime factorization - Prime knot

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