5 From Tangle Fractions to DNA 73[-2] [-1] [0] [1] [2]
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Fig. 5.3.The elementary rational tangles
Fig. 5.4.The Reidemeister moves
Examples of rational tangles are illustrated in the right-hand side of
Fig. 5.1 as well as in Figs. 5.8 and 5.17 below.
We study tangles up toisotopy. Two two-tangles,T,S,inB^3 are said to be
isotopic, denoted byT∼S, if they have identical configurations of their four
endpoints in the boundaryS^2 of the three-ball, and there is an ambient isotopy
of (B^3 ,T)to(B^3 ,S) that is the identity on the boundary (S^2 ,∂T)=(S^2 ,∂S).
An ambient isotopy can be imagined as a continuous deformation ofB^3 fixing
the four endpoints on the boundary sphere, and bringing one tangle to the
other without causing any self-intersections.
In terms of diagrams, Reidemeister [18] proved that the local moves on
diagrams illustrated in Fig. 5.4 capture combinatorially the notion of ambient
isotopy of knots, links and tangles in three-dimensional space. That is, if
two diagrams represent knots, links or tangles that are isotopic, then the one