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Efficient knot group identification as a tool for studying entanglements of polymers
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10.1063/1.2806928
/content/aip/journal/jcp/127/24/10.1063/1.2806928
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/24/10.1063/1.2806928

Figures

Image of FIG. 1.
FIG. 1.

An embedded graph whose knot group is analyzed in Table I.

Image of FIG. 2.
FIG. 2.

A two-component link whose knot group proves to be the free group of rank 2, indicating that it consists of two unlinked rings.

Image of FIG. 3.
FIG. 3.

The knot group of this graph proves to be the trivial group, which indicates that the two paths are not knotted.

Image of FIG. 4.
FIG. 4.

An oriented projection of the unknot. Each region of the projection plane possesses an integer, , the winding number, which counts the number of full 360° cycles that the knot makes around that region.

Image of FIG. 5.
FIG. 5.

The knot group of this graph proves to be the free group of rank 2.

Image of FIG. 6.
FIG. 6.

This graph has the same group as Fig. 5, the free group of rank 2.

Image of FIG. 7.
FIG. 7.

The knot groups of all these knots prove to be the free group of rank 1, which establishes that each is a projection of the unknot.

Image of FIG. 8.
FIG. 8.

Two-component links with isomorphic knot groups. The knot groups of (a) and (b) are isomorphic, as are the knot groups of (c), (d), and (e), as well as the knot groups of (f) and (g).

Tables

Generic image for table
Table I.

Manipulations on knot group representations of the graph in Fig. 1.

Generic image for table
Table II.

Manipulations on representations of the knot group of the graph in Fig. 2. The final representation indicates that the knot group of this graph is the free group of rank 2.

Generic image for table
Table III.

Manipulations on knot group representations for the graph in Fig. 3. The final representation indicates that the group of this graph is the free group of rank 0.

Generic image for table
Table IV.

Manipulations on Fig. 4, the “nasty” unknot.

Generic image for table
Table V.

Manipulations on Fig. 6.

Generic image for table
Table VI.

Application of the knot group procedure to self-avoiding rings (i.e., -polygons). Approximately 50 000 instances of self-avoiding rings on the simple cubic lattice were generated at each of the values of indicated. The knot group procedure was applied to each of these. This table indicates the frequency with which each of several different knot groups were identified. F1 indicates that the knot group was found to be the free group of rank 1. indicates that identification of the free group required rebounds. indicates that the ring was identified as the knot , etc.

Generic image for table
Table VII.

Performance of the knot group procedure in repeated trials on several different projections of the unknot. CPU times are for Pentium III processors.

Generic image for table
Table VIII.

The most frequently occurring knot group representation for each of the indicated knots or links.

Generic image for table
Table IX.

Performance statistics for the procedure. CPU times are for a Pentium III processor. is the mean crossing number at the beginning of the procedure.

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/content/aip/journal/jcp/127/24/10.1063/1.2806928
2007-12-26
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Efficient knot group identification as a tool for studying entanglements of polymers
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/24/10.1063/1.2806928
10.1063/1.2806928
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