Calculating the Secrets of Life Contributions of the Mathematical Sciences to Molecular Biology (1995) / Chapter Skim
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Pages 212-221
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From page 212... ...
In chemistry and biology a nontrivial link is called a catenane, from the Latin catoena for "chain," since the components of a catenane are topologically entangled with each other like the links in a chain. In this excursion, we will restrict attention to dimers, that is, links of two components, because dimers are the only links that turn up in topological enzymology experiments.
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From page 213... ...
In the definition of knot type, we insisted that the transformation that superimposes one knot on another must be onentation-preserving on the ambient space. This restriction allows us to detect a property of great biological significance: chiraTity.
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~\ Cat FIGURE 8.4 (a) Hopf link, (b i figure eight knot, (c)
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, and so on. The algebraic classification schemes for these homologous subfamilies can be used to describe and compute enzyme mechanisms in the topological enzymology protocol.
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From page 216... ...
Also, standard 4-plat diagrams (with the exception of the unknot < 1 > are minimal (Ernst and Sumners, 1987~. For in vitro topological enzymology, we can regard the enzyme mechanism as a machine that transforms 4-plats into other 4-plats.
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From page 217... ...
< 2,1,1 > figure eight knot, (c)
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From page 218... ...
Tangle theory is knot theory done inside a 3-ball with the ends of the strings firmly glued down. Tangles are usually represented by their projections, called tangle diagrams, onto the equatorial disk in the unit 3ball, as shown in Figure 8.6.
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A classification scheme for rational tangles is based on a standard form that is a minimal alternating diagram. The classifying vector for a rational tangle is an integer-entry vector (a~,a2,...,an)
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From page 220... ...
Two rational tangles are of the same type if and only if they have identical classifying vectors. Due to the requirement that | al | > ~ in the classifying vector convention for rational tangles, the corresponding tangle projection must have at least two nodes.
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From page 221... ...
Given these constructions, rational tangles are summands for 4-plats.
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