10 - The Linking Number Algorithm
Published online by Cambridge University Press: 19 August 2009
Summary
In Chapter 6, we discussed a topological invariant called the linking number and extended this invariant to simplicial complexes. In this chapter, we provide data structures and algorithms for computing the linking numbers of a filtration, using the canonical cycles and manifolds generated by the persistence algorithm. After motivating this computation, we describe the data structures and algorithms. We end this chapter by discussing an alternate definition of the linking number that may be helpful in understanding the topology of molecular structures.
Motivation
In the 1980s, it was shown that DNA, the molecular structure of the genetic code of all living organisms, can become knotted during replication (Adams, 1994). This finding initiated interest in knot theory among biologists and chemists for the detection, synthesis, and analysis of knotted molecules (Flapan, 2000). The impetus for this research is that molecules with nontrivial topological attributes often display exotic chemistry. Such attributes have been observed in currently known proteins. Taylor recently discovered a figure-of-eight knot in the structure of a plant protein by examining 3,440 proteins using a computer program (Taylor, 2000). Moreover, chemical self-assembly units are being used to create catenanes, chains of interlocking molecular rings, and rotaxanes, cyclic molecules threaded by linear molecules.
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- Information
- Topology for Computing , pp. 171 - 180Publisher: Cambridge University PressPrint publication year: 2005