Topology in Molecular Biology
70 L.H. Kauffman and S. Lambropoulou defined for the larger class of all 2-tangles. In this study we are interested in different ...
5 From Tangle Fractions to DNA 71 to alternating diagrams. The alternating form is used to obtain a canonical form for rational ...
72 L.H. Kauffman and S. Lambropoulou A two-tangle is an embedding of two arcs (homeomorphic to the interval [0,1]) and circles i ...
5 From Tangle Fractions to DNA 73 [-2] [-1] [0] [1] [2] , ,,, , ,,,, , ... ... ... ... [ ] [-1] _ [1] _^1 [2] (^1) 1 [-2] ^1 Fi ...
74 L.H. Kauffman and S. Lambropoulou flype t t t flype t ~ ~ Fig. 5.5.The flype moves diagram can be obtained from the other by ...
5 From Tangle Fractions to DNA 75 Tr T r = = -1/T , - = 1/T T S T S , , -T -T T+S T*S Ti T ~ i = Fig. 5.6.Addition, product and ...
76 L.H. Kauffman and S. Lambropoulou T T + = T T = Fig. 5.8.Creating new rational tangles T N N(T) T D D(T) T ~ T Fig. 5.9.T ...
5 From Tangle Fractions to DNA 77 D(T) interchange N(T) = [0] [ ] T T = Fig. 5.10.The [0]–[∞] interchange N(T)=N(T+ [0])−→N(T+[∞ ...
78 L.H. Kauffman and S. Lambropoulou Definition 2 Acontinued fraction in integer tanglesis an algebraic descrip- tion of a ratio ...
5 From Tangle Fractions to DNA 79 fora 1 ∈Z,a 2 ,...,an∈Z−{ 0 }andneven or odd. Thelengthof the contin- ued fraction is the numb ...
80 L.H. Kauffman and S. Lambropoulou Fig. 5.11.Reducing to the alternating form Recall that a tangle is alternating if and only ...
5 From Tangle Fractions to DNA 81 Proof (of Theorem 1).We have now assembled all the ingredients for the proof of Theorem 1. In ...
82 L.H. Kauffman and S. Lambropoulou where δ=−A^2 −A−^2. 3.〈K〉satisfies the following formulas 〈χ〉=A<>+A−^1 〈)(〉 〈χ〉=A−^1 ...
5 From Tangle Fractions to DNA 83 S = A = A -1 LL R R S L R Fig. 5.12.Bracket smoothings Define thestate summation,〈K〉, by the f ...
84 L.H. Kauffman and S. Lambropoulou : or : or ++ Fig. 5.13.Crossing signs and curls in Fig. 5.13. Note that the ...
5 From Tangle Fractions to DNA 85 knots and links, we can find astate summationformula for thebracket of the tangle, denoted〈T〉, ...
86 L.H. Kauffman and S. Lambropoulou Lemma 6.fracT(A)is an invariant of ambient isotopy for two-tangles. Proof.SincedT andnT are ...
5 From Tangle Fractions to DNA 87 We will letn(T)=nT( √ i)andd(T)=dT( √ i),so that F(T)=i n(T) d(T) . Lemma 8.The two-tangle fra ...
88 L.H. Kauffman and S. Lambropoulou S double resmoothing S ` Fig. 5.14.A double resmoothing [0] - [ ] interchange S S ` contrib ...
5 From Tangle Fractions to DNA 89 Atδ= 0 we also have: 〈N([0]〉 =0,〈D([0])〉 =1,〈N([∞])〉 =1,〈D([∞])〉=0,and so, the evaluations 3–5 ...
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