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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.
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Whats a rational tangle?
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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Whats a rational tangle?
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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Whats a rational tangle?
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.
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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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