KNOT THEORY

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KNOT THEORY
Take a piece of cord. Tie a knot in it and then
glue the loose ends together to form a closed
loop.
Unsolved problem how do we tell when two knots
are the same ?
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There is no foolproof way to tell if two knots
are the same or different.
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The left and right trefoil are different knots.
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The left and right trefoil are different knots.
The left and right figure 8 knots are the same.
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These knots are known as the Perko Pair.
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These knots are known as the Perko Pair. They
were thought to be different knots for over 75
years.
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The study of knots began in the late 19th
century. Lord Kelvin was very impressed by
Helmholtz's demonstrations of stable vortex
rings. He thought that atoms might be knots in
the aether. This idea was eventually
abandoned by physicists. Meanwhile...
mathematicians became intrigued by knots.
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A sample of knots and links. Mirror image knots
excluded. A link is two or more knots. All of
these knots are distinct.
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Knots in DNA
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Enzyme action on DNA is analyzed with tangle
equations.
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Knot theory is being used in the emerging field
of DNA nanotechnology. Novel chemical structures
invovlving knots.
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Knotting in catheters.
Knot theory in surgical simulators.
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Quantum foam loop quantum gravity.
String theory. Statistical mechanics. Quantum
field theory. Topology. Quantum computers
anyons. Braids. Ribbons. Tangles. Higher
dimensional knot theory (knotted surfaces).
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Knots turn up in a previously unexpected connecti
on with statistical mechanics. The connection was
found by Vaughan Jones in 1984.
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Knots appear in high energy physics via
Feynman diagrams, in string theory and knotted
surfaces.
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The three Reidemeister moves. They leave a knot
unchanged and help to find invariants.
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Knots as art.
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