Knots, Continued Fractions and DNA

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Knots, Continued Fractions and DNA
AiO Seminar Mathematics Friday 17-11,
1600-1700, Room P.014

• Roland van der Veen

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A knot is a circle embedded in space

• History of knot theory
• Rational tangles and continued fractions
• Classification of rational tangles and rational
  knots
• Application to DNA

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Peter Tait (1883)
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Method alternating knot diagrams
Tait Flyping Theorem (proven in 1990) Two
alternating diagrams give rise to the same knot
iff they are related by a sequence of flypes.

• Flype move

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John Conway (1970)

• Tangles

8
0

• Rational tangles
• Start with 0 and twist an odd number of times
  right, down, right, down, right,
• Start with 8, and twist an even number of times
  down, right, down, right,

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Conways Classification Theorem

• Let S and T be two rational tangles with twist
  sequences s1,, sm and t1,, tn.
• S and T are equal iff sm ,, s1 tn ,, t1

a ,b,c,d,e,f is the continued fraction
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Proof, Louis Kauffman (2003)

• Imitate the arithmetic of continued fractions
  with tangles

-T is T with all crossings reversed
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• A rational tangle can be written as its own
  continued fraction!

continued fraction 3,2 3
2,3
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• Every continued fraction has a unique canonical
  form a positive/negative continued fraction of
  odd length.

• Tangles with equal fractions are equal
• Bring the tangles into canonical form.
• The corresponding fractions are also in
  canonical form.
• The canonical form is unique for fractions, so
  the fractions are equal.

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Conversely Tangles with different fractions are
different.

• Bring the fractions into canonical continued
  fraction form.
• Bring the tangles into canonical form. The forms
  look the same!
• The corresponding diagrams are alternating, so
  they are related by flypes (Tait flyping
  theorem).
• Flypes do not change the fraction.
• The fractions are assumed to be different, so the
  tangles are not the same.

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Rational knots

• A rational knot is the closure of a rational
  tangle.

Notation cl(T)
cl(3)

• Theorem
• cl(p/q) and cl(p/q) are equal iff
• p p
• q q (mod p) or qq 1 (mod p)

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Application to DNA
X

• The DNA loop is twisted n times.
• The enzyme X replaces the tangle 0 by r.
• The result is the knot cl(10/7).
• Determine r without knowing n.

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Enzyme X

• cl(1/n r) the result is cl(10/7)
• nr 1 10 and
• either n 7 (mod 10) or 7n 1 (mod 10)
• nr 9, so n 1, 3, 9.
• The possibilities are n 3
• Enzyme X acts by replacing the 0 tangle by 3

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The infinite golden braid