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Figure 3.

(,) The ideal trivial -graph, obtained via PB-SONO, with /=5.12. (,) The ideal 3 entangled -graph isotope, with /=17.44, comparable with the energy for an ideal trefoil plus 1. (Online version in colour.)

Approximate ropelengths of many tight knotted θ -graphs can be estimated from related knots. The 3 1 θ -graph isotope is equivalent (by ambient isotopy) to a trefoil knot with a short additional edge; we expect its ropelength to be close to that of the ideal trefoil plus D units (the length of the extra edge), which translates to a ropelength of 16.38+1=17.38. In practice, we obtain a ropelength of 17.44 ( figure 3 c , d ). We note that the ideal conformation of the 3 1 θ -graph is not unique: the short connecting edge may be located at any point where different parts of the knot trajectory are in contact. Similarly, the 4 1 θ -graph embedding is the 4 1 (figure-eight) knot with an additional short edge. The ideal tight conformation, as shown in figure 4 a , d , therefore has a ropelength of 21.72+1=22.72 (the ideal figure-eight knot plus one). In practice, our implementation of PB-SONO yields L / D =22.77. In this case too, the tight configuration is not unique, as the location of the short edge has some flexibility, giving many ideal configurations equal (minimal) length.

Figure 4.

Ideal conformations, obtained using the PB-SONO algorithm, of three isotopes of the -graph: 4, 5 and 5 (named after Litherland's enumeration of -graph isotopes). Tight ropelengths are listed for each ideal embedding. (Online version in colour.)

The 5 2 θ -graph isotope has the next longest ropelength when tightened. This isotope contains a trefoil, with its additional edge wrapping around outside the trefoil. The presence of this edge necessarily distorts the ideal conformation away from the ideal trefoil embedding, as shown in figure 4 b , e . Here, L / D =24.24. The 5 3 θ -graph isotope contains a cinquefoil knot with an additional short edge. When tightened ( figure 4 c , f ), L / D =24.79, comparable with the cinquefoil (23.67) plus one (for the edge). Slight deviations from the ideal knot trajectory (kinks) are seen at the vertices of the graph, which cause the difference in the values. These kinks are an artefact of the computational process.

The 5 1 θ -graph is a ravel [ 17 ], with no knotted or linked cycles. This isotope is the simplest example of a ravel. (An organo-metallic molecule containing this specific entanglement has been synthesized [ 18 ].) The ideal conformation ( figure 5 c , d ) has L / D =25.08 and exhibits threefold symmetry. Intermediate configurations of the simulation are shown in figure 5 a , b ; the symmetry of the ideal configuration emerges regardless of the asymmetry of the starting configuration.

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