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Knots, Continued Fractions and DNA
Roland van der Veen
AiO Seminar Mathematics: Friday 17-11, 16:00-17:00, Room P.014
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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)
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Tait.html -
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Method: alternating knot diagrams
Flype move:
Tait Flyping Theorem (proven in 1990):
Two alternating diagrams give rise to the
same knot iff they are related by a sequence
offlypes.
T
T
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John Conway (1970)
Tangles
Rational tangles:Start with 0 and twist an oddnumber of times:
right, down, right, down, right,
Start with ,and twist an even number of times:down, right, down, right,
0 T
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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] = [t
n,, t
1]
[a ,b,c,d,e,f] is the
continued fraction:
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Proof, Louis Kauffman (2003)
Imitate the arithmetic of continued fractionswith tangles:
S T+ = S T
-T is T with all crossings reversed
3 -2 3 -2+ = 1
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A rational tangle can be written as its own
continued fraction!
T
= T-1
22
1
2,3
continued fraction: [3,2] = 3 +12
3 +21 = =
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 incanonical 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.
T
Theorem:
cl(p/q) and cl(p/q) are equal iff
1. p = p2. q = q (mod p) or qq = 1 (mod p)
Notation: cl(T) = cl(3) =
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Application to DNA
X
1.The DNA loop is twisted n times.2.The enzyme X replaces the tangle 0 by r.
3.The result is the knot cl(10/7).
4.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