Roland van der Veen- Knots, Continued Fractions and DNA

8/3/2019 Roland van der Veen- Knots, Continued Fractions and DNA

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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

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Roland van der Veen- Knots, Continued Fractions and DNA