Erica Shannon and Jennifer Townsend- Rational Tangles

8/3/2019 Erica Shannon and Jennifer Townsend- Rational Tangles

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Introduction Operations Equivalence Continued Fractions The Bracket Polynomial Applications Sources

Whats a rational tangle?

Tangles

Definition

A tangle is analogous to a link except that it has free ends whichare restricted to the boundary of a ball or box.

Figure: A tangle and a non-tangle.

Erica Shannon and Jennifer Townsend

Rational Tangles
http://find/http://goback/


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Rational tangles

Definition

A rational tangle is a tangle made of two arcs that can beunwound completely while keeping the endpoints of the arcs on the

boundary of the ball.

Figure: Various tangles, mostly rational.


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Construction of rational tangles

To make a rational tangle, take two arcs in a ball, choose any pairof endpoints and twist them, then repeat this a finite number oftimes with any pairs of endpoints.

Figure: Construction of a rational tangle.


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Canonical Form

Definition

A rational tangle is said to be in canonical form provided that thetangle is alternating.

Figure: A tangle in canonical form and a tangle decidedly not incanonical form.


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O C S


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Monomial Operations

Mirror Image

Figure: Mirror image of a rational tangle, denoted 1/a.


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I d i O i E i l C i d F i Th B k P l i l A li i S


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Monomial Operations

Flips and Flypes

Figure: A horizontal flip.

Figure: A flype.


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Monomial Operations

Numerator Closure and Denominator Closure

Definition

The numerator closure of a tangle is the knot/link made byconnecting the North endpoints to each other and the Southendpoints to each other. Similarly, the denominator closure ismade by connecting the East endpoints and the West endpoints.

Figure: Numerator closure and denominator closure.


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Binomial Operations

Addition

Figure: Tangle sum.


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Binomial Operations

Tangle Product

The tangle product ab = a + b.

Figure: Tangle product.


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Binomial Operations

Ramification

The ramification a, b = a + b.

Figure: Tangle ramification.


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p q y pp

Definition

Equivalence of Rational Tangles

DefinitionTwo tangles are equivalent if they can be deformed to each otherwithout moving the endpoints. Allowed deformation moves areanalogous to Reidemeister moves.


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Applications

Nifty Equivalence Properties (I)

Figure: Two tangles with the same canonical form.

Two tangles are equivalent iff they can be deformed to the samecanonical form.


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Applications

Nifty Equivalence Properties (II)

Any rational tangle is equivalent to an alternating tangle.

Figure: A rational tangle with an extra pink kink and its equivalentalternating tangle.


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Calculation

Vector Notation for Tangles

A tangle can be represented in vector form (a1, a2, . . . , an) whereai is the number of consecutive twists of a pair of endpoints.

Figure: Rational tangles and evolution of their associated vectors.

Erica Shannon and Jennifer TownsendRational Tangles



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Calculation

Continued Fractions of Tangles

The continued fraction of a basic tangle (denoted F(t))

corresponds to this set (a0, a1,..., an) of twists as

F(t) =1

an +1

an1+1

an2+1...




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Calculation

An Example

Figure: A rational tangle and its associated fraction.




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Properties

Properties of Continued Fractions of Tangles

The continued fraction of a rational tangle T is related to itsinverse and mirror image as follows

F(1/T)=1/F(T)

F(-T)=-F(T)

Flypes do not affect a tangles continued fraction.

The continued fraction is a rational tangle invariant.




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Conways Theorem

Conways1 Theorem

TheoremIf T1 and T2 are basic tangles, then F(T1) = F(T2) implies thatT1 is ambient isotopic to T2.

1Squaredancer Extraordinaire




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Calculation

The Bracket Polynomial for Rational Tangles

Weve already defined a bracket polynomial for links and knots.The bracket polynomial for tangles is similar . . .

J

= A

+ A1

||. . . but we end up with a Laurent polynomial with unresolved terms|| and =.




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Calculation

The Bracket Polynomial for Rational Tangles

So starting with

J = A + A1||

we define (T) and (T) for some tangle T by

T = (T)|| + (T)=.

So for J, we have

(J) = A1

and(J) = A.




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Properties

A Bracket Polynomial , Theorem!

Theorem

RT(A) =(T)

(T)

is an invariant of tangles.




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Magic

In Which Everything is Illuminated!

Now take RT(A) and replace A, for magical reasons, with

i.




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Magic

In Which Everything is Illuminated!

Now take RT(A) and replace A, for magical reasons, with

i.

Theorem(Goldman and Kauffman)

F(T) = iRT(

i).

Remember, RT(A) =(T)

(T) .




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DNA as a Rational Tangle

Purely Applicable: tangled DNA

Figure: DNA is really a series of a zillion tangles, many of which arerational!




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Sources

C. Cerf, A Note on the Tangle Model for DNA Recombination, Bulletin of Mathematical Biology (1998), pp 67-78.

I.K. Darcy, A rational tangle primer, Available: http://www.knotplot.com.

J.R. Goldman, L.H. Kauffman, Rational tangles, Adv. in Appl. Math. 18 (1997), no. 3, 300-332.

L.H. Kauffman and S. Lambropoulou. Knotting, Linking, and Folding Geometric Objects in R3. AMS SpecialSession on Knotting and Unknotting. Las Vegas, Nevada. 21 Apr. (2001), pp 223-260.

J.C. Misra and S. Mukherjee, Mathematical Modelling of DNA Knots and Links. Biomathematics: modelling andsimulation. World Scientific (2006), pp 195-224.

Wikipedia.

Y. Saka, M. Vazquez, TangleSolve: topological analysis of site-specific recombination, Bioinformatics, Available:http://bio.math.berkeley.edu/TangleSolve/tmodel/tmodel.html, (2002) 18:1011-1012


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Erica Shannon and Jennifer Townsend- Rational Tangles