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Knots and links made of one-dimensional lines embedded in a three-dimensional space are archetypes of topological intricacies. Their classification involves the number of crossings which are, by definition, immune against the rewiring. An exhaustive and accessible review on this topics was published in Reports on Progress in Physics by I. Smalyukh.
Tangles
the generic concept introduced in the theory of knots by John Conway
Louis H. Kauffmann
“John H. Conway introduced the notion of tangle and defined the fraction of a rational tangle using the continued fraction form of the tangle and the Alexander polynomial of knots.
Conway was the first to observe the extraordinary interplay between the elementary number theory of fractions and continued fractions and the topology of rational tangles and rational knots and links”.
Conway's construction of a rational tangle using the continued fraction of 29/12
We have pointed out that stable helical tangles can be wound up from cholesteric dislocations. For this reason, cholesteric dislocations can be tied into stable knots and links.
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